I haven't been back to Vancouver for a couple years, but if you're downtown at Hastings St. and Main St. anyway, walk south along Main St. for a few blocks and you're at Chinatown -- lots of history and character! Downtown's Chinatown Nightmarket runs weekends in the summer after 6pm to 11pm, giving the area some nighttime activity.
And while the scenery at Hastings St. and Main St. is not all roses, most of the times the residents or occupants (questionable or not) will not bother you (at least during daytime hours!), even though their appearance and demeanor may make you feel uncomfortable! One year, my sister and her young family wandered into that neighbourhood (from Chinatown, right into the thick of things!) and nothing happened! Another year, we (my wife and I) met a group of friends (fellow Saskatchewanians) at a restaurant on East Hastings -- we drove there, our friends were unfamiliar with the downtown layout and walked along East Hastings from their downtown hotel, crossing Main St. on the way and also did not encounter problems! (probably because the "polite Canadian" is ingrained into most of the locals!

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Some of our favourite places to visit are Granville Island, Lonsdale and Westminster Quays (public markets like Granville Island, accessible by Seabus or Skytrain), UBC, of course Stanley Park, and White Rock (boardwalk area).
If you're into hiking: Capilano Suspension Bridge, or (for free) Lynn Canyon, Lighthouse Park, Grouse Mountain and the Grouse Grind, even Whistler by car or bus! Or take a walk along any part(s) of the Seawall that starts from Stanley Park, and continues alongside the neighbourhoods of English Bay, trendy Yaletown, and fairly brand new False Creek Southeast (former Olympic Village). Beware: for hiking/walking, daytime temperatures in the summer can be quite warm, so start early (or if you're just staying within the city, you can start later) and/or bring enough drinks to keep hydrated! Oh yeah! There's the 2010 Winter Olympics Cauldron in Coal Harbour, north side of downtown! There's also the short ferry rides within False Creek on the Aquabus (stops at Science Centre, Granville Island, Yaletown, Olympic Village, etc.). Or even book a harbour boat tour of Burrard Inlet. You won't be disappointed!
If you're into shopping or browsing: Robson St., Davie St., Broadway Ave., Pacific Centre Mall, Oakridge Mall in or near downtown area, Metropolis at Metrotown Mall in Burnaby, Aberdeen Centre (Asian supermall) in Richmond, Ikea (their restaurant for cheap food), and if you're into atmosphere, Richmond's Chinatown and Summertime nightmarkets! If you like quaint, neighbourhood shopping/browsing, there's Kitsilano and Kerrisdale neighbourhoods! Most of these places are accessible by Skytrain or transit, too, if not by car.
There's a myriad of restaurants ranging from expensive places like Hawksworth in downtown; to popular places like numerous locations of Earls, Cactus Club, Milestones; burger places like White Spot, Red Robin, Fatburger; tons of various Chinese, Korean, Japanese, Indian and other ethnic restaurants, esp. along Robson St., Davie St., Broadway Ave., Kingsway, and Main St.; and lots of food carts/trucks.
On your way back, if you're driving and you haven't been there, I recommend visiting Seattle, esp. Pike Place Market. I'm still waiting for P.F. Chang's chain of restaurants (we ate at the one in Lynnwood, near Seattle) to establish a Vancouver location, like they did in Toronto and Montreal already!
Welcome to Canada and have a good time!