There are a number of hostels in Vancouver but I do recommend booking ahead as I know they can get crowded especially if you want a private room. Hostelling Int'l operates 2 hostels that I know of one is located downtown in the Westend, the other is located at Jericho beach in the Point Grey neighbourhood of Vancouver.
The Westend hostel is close to the downtown nightspot and is about a half hour walk away from Stanley Park and a five minute walk for the trendy shops on Robson street. The Jericho hostel is a little quieter and its biggest drawback is that you are not directly on any bus route so if you go out at night and party late into the wee hours you're cabbing home.
There is also the "Backpackers Hostel" on Granville street right downtown which could be to your liking as there is a pretty hopping bar downstairs.
As for what to do. Well, you have to go and see Stanley Park it is simply mandatory. It is the second biggest park in North America after Central Park and has loads of amenities such as the acquarium with killer whales a petting zoo, the minature train ride and beautiful views throughout. If you walk around the sea wall you'll see great vistas of the city. Climb Grouse Mountain or take the Gondola for more great views! UBC once again has awesome views of vancouver Island snow capped mountains and the Pacific Ocean. Travel to commercial Drive. for the best coffee in the city and some good ethnic cooking eveyting form Ethiopean to Italian. Chinatown is also pretty cool and is the second biggest Chinatown in North America after San Francisco's. I also recommend you go to Steveston which is in Richmond for some good fish and chips. It is a quaint little riverside "town". Finally, no trip to Vancouver is complete without gorging oneself on sushi. we have some of the best sushi on hte planet and it is cheap compared to almost anywhere else. For good all you can eat Sushi I recommend Tampopo on Denman. However, if you are feeling more upscale just look around you are bound to stumble onto something good.