g'day, sydney here.
walk around the rocks, harbour bridge, opera house, botanic gardens, darling harbour, catch a ferry to manly, grab a few beers, eat fish and chips on the beach. you can spend a weekend wandering around the harbour and galleries / museum. taronga zoo is also worth a day. its still summer so head to the beach. bondi is the world famous touristy one but you can head up or down the coast and find your own stretch of coastline if you are so inclined (trains go up and down the coast or rent a car). plenty of snorkelling, surfing, and other water activities. scuba diving is ok but you really want to head up to the barrier reef for that. to get out of town, the south coast is pretty nice down kiama way.
for bushwalking, sydney is surrounded by national parks north, south and west and the ocean on the east. close by and easily accessible is the blue mountains (katoomba to start with) and theres lots of bushwalking around there. people get lost there and have to be rescued all the time. we had a case last week where a kid was lost in the bush for 3 weeks and found one kilometer from his house in bushland. anyway, you won't have any problem finding bushwalks around sydney, they are well travelled.
food is ok. pick a country and you'll find its restaurant. the centre of the city has pretty much been taken over by asians so standard options are some form of asian or pub fare (steaks and burgers and schitznels etc). plenty of posh restaurants too but thats up to your budget and tastes.
a lot depends on your age and interests. where are you staying in sydney? renting a car or public transport? you want to party or do grown up stuff? our trains are ok to get out of town and buses are regular and will shuttle you around the city but we don't have a metro or any sort of super efficient public transport like that.
i've been to melbourne a lot and it's a cool city, most of my guilds have lived there so i've been down for lots of guild piss ups. the women are hotter. it has a good cafe / restaurant vibe and the public transport is decent if confusing. can't give you much more outside what i do when i'm there which is visit galleries and getting pissed in st. kilda with guildies and talk about games.
i'm heading off to san diego for 6 weeks for work so i was planning on starting a thread like this myself. have a good trip and if you need more info let me know.