Come on you guys. The Original Poster should be advised to stay in his car and to drive mindfully when checking out East Hastings. It's Vancouver's skid road. I believe the poorest postal code in Canada. The locals pretty much stick to themselves and it has it's own community feel but it's got some shabby spots. Some alleys to avoid. Many stagger/cross the road without looking so try not to hit anyone. It always fascinated the boys when their bus came into town and they drove down Hastings. Try to spot the dealer was a favourite game. You could get a cab to take you for a little whirl around the neighbourhood.
http://www.lonelyplanet.com/canada/vancouver/things-to-do/top-things-to-do-in-vancouver Some great ideas here.
Gastown is a street or two north of Hastings and has a touristy vibe and more upscale restaurants and trendy galleries. Some interesting shops i.e.:
http://cavaliergastown.com/about-us-cavalier-gastown. Seabrooke's younger brother is part owner. Strangely upscale within blocks of cardboard beds in doorways. China Town is also a neighbour. Some traditional Chinese apothecary's are interesting. Also authentic food and bakeries.
How old are you? Maybe your style is Robson Street, YaleTown, Granville, Alberni St., Denman Street and vicinity for some shopping, clubs and restaurants. Skytrain stops at Metrotown, Burnaby one of the largest malls in BC and 2nd largest in Canada.
Stanley Park and it's sea wall walk/roller skate/bike is popular. It's Aquarium is a good spot if its raining. UBC has a great Museum of Anthropology.
For a cheap overview of the City I'd take the sky train
http://www.translink.ca/en/Schedules-and-Maps/SkyTrain.aspx east all the way over the mighty Fraser River into Surrey with a stop at MetroTown in Burnaby. Could also take the sky train to/from Waterfront it's terminus by Canada Place/Convention centre with a stop there for drinks or just to walk around. The Olympic torch is near by but don't think its lit.? Take the The Seabus ferry across Burrard Inlet to North Van. and look back at Vancouver. Hope for good weather for that trip and its quite beautiful. Stanley Park to the west. Working ships and the harbour cranes. Float planes taking off and landing. One of my favourite things to do. Check the trip time so you leave yourself enough time and also try to avoid rush hour. Or not. Also a unique experience.
No matter what you decide you'll find something interesting to do.
Enjoy.