General
This cosmopolitan metropolis with population of 4.5 million inhabitants is unique. You won't be bored here. Latitudinal stretch is around 80km.
Climate has an positive impact on your feeling - not too hot summers and relatively mild winter (May-August). There are sometimes hot days, but the breeze from the ocean makes you feel perfectly here, the average temperature in January (summer) is 25˚C. Only in the winter we'll have to put some insulation layer. On cooler days the temperature falls at mornings below 10˚C.
Atractions
In the city there are many free events. The most of that is during festival in Christmas and lasts until the end of January. During this period something happen every day, is too much to describe it, so let me write only the most interesting one of annual events.
26 December in the city harbour starts sail regatta Sydney - Hobart, where the Watson cliffs are ideal spot to watch how more than 100 sailing-boats turn into the open ocean.
Australian Day is on 26 January - old car exhibition on Macquarie St, boat parade and fireworks in Darling Harbour, music concerts in Hyde Park, some museums free of charge.
Until mid-February every year open air cinema show movies in the Botanic Garden and Centennial Park.
Park
Domain held a free event on Saturday evenings in January and February, when thousands of people sit on blankets with wine and food, watching and listening for example short films, opera, carols, jazz or symphony. List of events on Wikipedia.
The first Saturday in March "Mardi Grass" is held, where homosexuals (San Francisco is their largest agglomeration in the world, Sydney is second) show the parade in their district along Oxford St.
St Patrick Day, 17 March, is a chance to drink a beer with the Irish who get dressed in green.
In the Olympic Park, a week before Easter, Royal Easter Show is visited by several million people annually. It is about livestock and domestic animals, it doesn't sound great, but the unconventional method of presentation makes it impossible to be bored there. In addition to a number of interesting exhibitions, we can see how to shear a sheep, try to milk a cow, watch the dogs competition or various other trainings. They also do rodeo and stuntman show.
In October in Hyde Park you can see photographs about Sydney daily life.
November is inviting for car exhibition, and "Sculpture by the sea" between Bondi and Tamarama Beach.
The first Tuesday in November is a Melbourne Cup horse race. In Sydney at 3 pm all pubs are full with screaming people wishing the horse they bet to win.
These are just some of the fixed points of the program, many events take place suddenly, so we have to constantly search what is going on. World Press Photo usually is held in State Library in Domain. In Darling Harbor entertainment centre each week is a free public festival, shows or exhibitions. You can see concerts, dances from different parts of world, or admire acrobatics on water skis. There are also Chineese Gardens, Wildlife Park, Aquarium (35 AUD. In the summer at least once a week we can see amazing fireworks.
There are not many interesting old monuments or buildings in the young city like Sydney, but The Rocks district has some nice atmosphere. The main attraction of the city is a futuristic building of the opera. The prices for the concert starts from about 50 AUD, but the opera tour is quite interesting too.
You can climb the Harbour Bridge from 263 AUD. However, for a similar interesting city lookout you can climb one of the bridge towers (pylon) for just 15 AUD. For even better lookout you get catch a lift to the top of TV tower (25 AUD).
Whale watching can be done by boat (3 hours, 75 AUD, between May and September). We can have a close up look at those fantastic humpbacks.
In the information centre ask for a booklet about Sydney. There are vouchers with 20% discounts on admissions to selected attractions.
Here are some other attractions in the city:
- City Sightseeing decker bus hop-on hop-off tour around the city with driver commentary, 50 AUD per 24 hour ticket (nothing special)
- Ocean World Manlywhere you can dive with sharks
- Jetboat ride next to the opera, from 69 AUD
- Capitan Cook Cruise on the boat at Sydney Port, from 45 AUD
- Sydney Observatory (nothing special)
- skydive with landing on the beach from 265 AUD
- Party Bus (ride from pub to pub) or Party Boat (dancing with music and alcohol), from 40 AUD
And museums:
- Sydney
- Powerhouse
- Maritime
- Art Galery (free)
- Contemporary Art (free)
- Australian (specially worth during the annual photographic exhibition Wildlife Photography)
Describing the events, attractions, etc. it isn't possible to reflect Sydney atmosphere, which is incredible positive. People are very easy going and helpful, and you can hear almost everywhere the face with a smile saying "how is going?" and "no worries mate". There are some good websites to find out about oncoming events www.sydney.citysearch.com.au, www.cityofsydney.nsw.gov.au.
Activities
A very popular activities form during the weekends are picnics with BBQ in parks or on the beach, such as Bondi, Bronte and Coogee. The whole families and friends come there to sit together, eat, drink, chat and play the ball. There are a lot of choice of places equipped with BBQ stoves for coins or buttons. Most beaches are close distance from the City, just 20-40 minutes ride by city bus. Once on the beach, we have the opportunity to spend an even more active day. The perfect temperature of water is ideal for swim, surf, run on the sand. The beaches are guarded by lifeguards, safe from shark attacks, and deadly jellyfish and crocodiles are a few hundred miles to the north. In winter the ocean is still suitable for swimming, but only for warm people. Sometimes it seems comic situation when I see at the same time swimmer and watching him person in jacket, cap, gloves and scarf. During the summer do not forget to use protective UV filters, Australia is the leading country in terms of skin cancer.
Virtually all beaches are occupied by surfers. You may also learn how to ride on the waves in local beach schools. One day surfing lesson costs less than $100 (4 hours in water). Very beautiful and interesting are the cliff walks along the coast of the Pacific (Tasman Sea), where the total length of the Sydney coast is over 100 km, and the most interesting hiking trails around the city can be found in the booklet "Go walkabout". If you have only one day, I suggest 3-hour walk from Bondi to Coogee. In the City centre you can also relax in the nice scenery - free Botanic Garden, Hyde or Domain Park offer a peace among the green vegetation. I like the sign at the entrance: "Please walk on the grass, hug the trees etc". Australians are integrated with the nature.
Aussie love sport as that can be seen at every step. Number one is the rugby, they play it anywhere. Almost all pubs have TV with sporting events. Quite popular is also cricket, Australian football, tennis and swimming, and slowly 
soccer. In addition, cyclists, runners, triathlon athletes, skaters, horse riders and other athletes can be found in many parks, especially the large Centennial Park, which is an ideal place for any sport activity. Golf is played usually on the coastal areas.
In the second Sunday in August Sydney is hosting the biggest annual run race in the world, City to Surf, where around 70 thousand competitors struggling with their own weaknesses on this 14 km route from the city centre to Bondi beach. Multiplied the distance by each runner we will receive an impressive result of run around the earth as many as 24 times! This is really a good experience and anyone who is defeated himself is the winner.
In September marathon race is held (hilly - do not count on a good time), and also 10 km run via Harbor Bridge overlooking the opera.
Other interesting sport event is swimming, when at the end of summer at Bondi and Manly beach more than 2,000 professional and amateur swimmers are going to race in the ocean for a distance of about 2 km.
In March triathlon competition is held next to the opera, where athletes struggle with their weaknesses.
General information
You can take your alcohol to cinema or restaurant, just look for advertisement BYO (bring your own). Shops which sell alcohol are called "bottle shop" and are closed at midnight. Later on we can drink only in bars or night clubs. The largest group of them are close to Oxford Street and King's Cross. The first one is mainly dominated by homosexuals, while the latter one by backpackers, tourists, prostitutes and drug addicts. Very popular fun district is also City, Chinatown and Darling Harbor. Generally Sydney is very safe city, you can walk at night alone.
Weekend around the city
You can spend a weekend not only in the city, as many nearby national parks just waiting for a visit.
Blue Mountains offer many attractions (2 hours drive, 100 km west, on Sundays Opal Card ticket cost 2.70 AUD). Organized one day tours starts from 60 AUD.
I suggest to stop at Wentworth Falls and take a two hour walk to the waterfall and continue the trail in the middle of the cliff (National Pass). The path descend steeply at first, then it runs 3 km in the relatively flat terrain, then you have to climb back to reach the cliff rim. Anyway, the views are worth the effort.
The most touristy place is Echo Point in Katoomba, the icon of the Blue Mountains - Three Sisters. Here we can take some walks too, in the lower or upper parts of the park. In addition to interesting tracks, we can ride the steepest railway in the world (Scenic Railway, the slope of 45%), or admire the scenery from the windows of the cable car (Scenic Skyway).
If you have a car, it's worth to drive up to lovely locations near Blackheath - Govetts Lookout, Evans Lookout, and especially my favourite - Pulpit Rock.
I described only the most popular places. It’s possible to see them even in a single day. But really, the Blue Mountains offer much more (stretch over 200 km). 
There are few national parks next to each other - Blue Mountains, Wollemi, Gardens of Stone and Kanangra Boyd. Their wildernesses invite trekkers, rock climbers, abseiling lovers, canoeing amateurs etc. For those less active, but still nature lovers, park offers organized, quiet and simple campsites with car access.
Many caves around are open to visitors (the most popular one is Jenolan Caves). On the way from Sydney to the Blue Mountains, you can see as well Featherdale park with animals (personally I do not like parks like that one, but if someone does not have occasion to see animals in the wild, it can do it here, admission 32 AUD).
For me personally the most beautiful park near Sydney is Royal, the second oldest national park in the world (after Yellowstone). Catch a train to Cronulla, and then a ferry to Bundeena (return trip around 25 AUD). After getting out from ferry I would recommend turn left to the beach. At the beach end, climb wooden stairs, walk down the street and turn to the next beach (Jibbon Beach). At the end of it the trail begins in a small bush. From here you can go along beautiful cliff coast. From time to time the trail runs inland, but I suggest sticking to the cliff – better views. After about 3 hours walk from the ferry you will reach paths intersection. The right turn leads to Bundeena (30 minutes walk), and if we go straight along the cliff for another 30 minutes, we will reach beautiful, white, perfectly even walls rock which looks like Camembert cheese (Wedding Rock). I really like this place. Now we have two options - return the shortest route to the ferry (an hour away), or continue on via Marley 
Beach to Wattamolla. There is a parking lot, so we can try hitchhiking to the train station. A few more hours away is another parking lot full of surfer’s cars, called Gari Beach. Just behind it there is South Era's Campground (the permit must be applied in advance). The Era is a couple of hours from Otford, where we can catch a train. The entire route between Bundeena and Otford is 26 km, ideal for 2 days. If you have a little luck you can see dolphins from the cliffs. And between May and October, whales too.
For a one day trip I suggest to go to the north tip of Sydney (catch a ferry from Palm Beach), to Ku - Ring - Gai Chase NP with beautiful colourful bays.
If we have a full day (or easy going two days), it is worth visiting Port Stephens with high sand dunes. Driving 4WD off-road along the beach and dunes, and have some rides downhill on a sand board. There are also boat trips for dolphins watching. Organised day trip from Sydney costs 170 AUD.
If we are wine lovers, the Hunter Valley invites you to their vineyards. However, it can’t be compare to French ones. Free wine, cheese or chocolate tastings attract volunteers. The only problem is who will be driving on the way back? Tours prices from 95 AUD.
Jervis Bay is famous for its very white and fine sand. In addition to turquoise water, this is very pleasant place. Campsites are located next to the sea. There are also nice trails, but we encountered snakes twice already. You just have to look at your feet and everything will be okay.
It remains to visit the capital. However, Canberra is nothing special for me. If not the annual flower exhibition, there is no reason to go. Although there is very original parliament building, built inside a hill.
One week in Sydney
If I had only one week in Sydney, I would spend a time as follows:
1) City Centre, Opera House, Botanical Gardens, the view from the Harbour Bridge, The Rocks, Cathedral, Hyde Park
2) Costal walk from Bondi to Coogee beach, 7 km long along the ocean, duration 3 hours plus getting to the beach by bus. If you like hiking a lot, I would start at Watson Bay, walk a bit and catch the bus to Bondi.
3) Ferry cruise from Circular Quey to Manly. Over there walk to Shelly Beach and North Head.
4) Full day excursion to the Royal NP, a costal walk.
5) If anything left from the first day. If not, do the same at night time. Look what's going on, what festival takes place. Depending on your needs go to a museum, aquarium, etc.
6) Full day excursion to the Blue Mountains
7) Chinatown, Chinese gardens (if you weren’t in Asia), buying souvenirs in Peddy's Market. Evening stroll at Kings Cross and along Oxford St.
Getting around
Sydney Transport - the city is divided into zones and you are paying for a ticket depends on it. If you live 15 km from city center you will pay weekly 44 AUD for return train ticket. Opal Card structure is a bit complicated, so I suggest you to check all conditions on their website and find out more about 30% discount for travelling off-peak hours, 60 minutes transfer mode, 8 travels a week and then ride for free, Sunday fee of 2.50 AUD!!! For buses in the city centre you won’t be able to buy a ticket from the driver between Mon-Fri, 7 am till 7 pm.
Very good and clear information can be found on www.cityrail.info for trains, and www.sydneybuses.info for buses.
When it comes to arriving to Sydney. Despite the fact that the airport is located near the city center (10 km), travelling over there is expensive. Taxi costs more than 50 AUD, train costs 19 AUD. Another alternative is to find a "shuttle" to one of the hostels, starts from 14 AUD. There is also a bus (No. 400), but this service runs between Bondi Junction and Burwood - and to get to the city center, you have to change a bus, whole trip takes over 90 minutes (15 minutes by train).
There is one more way to get to the city cheaper - leave the international airport terminal 1, go straight through the parking lot to the bridge (5 minutes walk), which pass to 
the west side of the channel. Immediately afterwords crossing the bridge turn right and go along the same channel to the north, cross the main road Princess Hwy and now you have only a few hundred meters to the Wolli Creek train station (from the airport 30 minutes walk). The ticket from here to the city center costs only 5.40 AUD. As regards to the domestic airport, walk from any terminal in the same direction as the cars go. You get to the main road with traffic lights - cross it and continue march north along O'Riordan St for about 500m, then turn left in Bourke Rd. Now, just a few hundred meters walk you find on the right side Mascot metro station (about 25 minutes walk from the airport). The ticket to the city center costs 4.40 AUD.
Hitchhiking out of Sydney -
If you going south catch a suburban train to Campbelltown, and from there walk following the signs for Hume HWY, leading to Melbourne.
Going north catch a train to Wahroonga and go to the "motorway" starting from the turn of the Pacific Hwy.
Leaving Sydney to the west, it is best to catch the train to Katoomba, and go on foot about 1 km along the main road to the west.