I always find it really hard to suggest things for people to do in Melbourne despite living here most of my life. I'm not really up on the sort of 'touristy' activities that are available, but I'll try and give you some ideas.
The botanic gardens in the city are really beautiful, and during the warmer months it's a great spot to sit with friends.
There's several different suburbs of the city with really cool and unique atmospheres, depending on what you like. St Kilda is a beachfront suburb with bars/clubs, might be worth a look, I personally prefer South or Port Melbourne beaches, and Clarendon St in Sth Melbourne is a lively shopping strip with bars/shops all the way upto the casino.
Brunswick or Fitzroy have your student, hipster sort of vibe, with an eclectic selection of shops and plenty of bars/clubs often with a less mainstream focus.
Of course, I also have to recommend Footscray, particularly for the range of exotic restaurants - mainly Asian and African.
If you're able to travel a bit, up near Belgrave is close to some national parks, gardens and forests. I love the rhododendron gardens nearby in Olinda, but it depends what kind of thing you enjoy doing. These little towns/suburbs can also be very interesting, with lots of coffee and devonshire tea shops, antiques, and natural/organic/spiritual type shops.
As others have mentioned street art is really big here, Hosier Lane in the city is one well known street for it and I think there's still a street art gallery there - but in Melbourne, you can walk down any laneway and see some amazing art, or take any train to the end of the line to check out the pieces on the way. The Hurstbridge line is my favourite for graffiti.
As for clubs, Melbourne has a really awesome nightlife. It depends on what kind of thing you're looking for, and what your musical tastes are. Apart from the city, other areas for clubs are Chapel St in Prahran, Glenferrie Rd in Hawthorn, St Kilda or Fitzroy. I'd recommend checking out Cherry Bar for 'maximum rock n' roll', it's on AC/DC lane in the city - which incidentally was renamed because of Cherry Bar. Revolver on Chapel St is trashy and fun and pretty much a Melbourne institution, it used to stay open from Friday night til Sunday night non stop, though I don't know if it still does. My Aeon on Sydney Rd Brunswick often has some mad psy trance nights/mornings. There's also hard trance clubs, metal clubs, live music venues, quiet little bars and mainstream house/rnb clubs with strict dress codes and you go to be seen. The casino might be worth a look if you like a gamble, or even if not, it's also just a big complex with high end shops, clubs, and restaurants.
It'd be easier to recommend some specific clubs/bars, and other possible activities if you give some idea of what you're into
