Vancouver, as you may have noticed, is not the best city in which to ride, at least on the street. However, there are more trails than i could possibly list, and lots of good rides just outside of the city.
I've lived here for about a year and these are some of my favorites

Some gentle trails: the Richmond seashore
stanley park seawall
Pacific spirit park

Road riding: Stanley Park Drive
around UBC
Airport circuit / Iona spit
River road, Richmond

For serious off-road, go to the north shore: Mt. Cypress and Seymour both have them. Whistler has super downhill.(for a real challenge, ride to the top of Mt Seymour parkway or Cypress Bowl road ; they're each about a 3000ft climb).

Enjoy!