I am new to BC so I haven't tried any of these trails my self but here are a few to consider.
The TransCanada trail is now complete, recent articles indicate that the Myra Canyon section from Kelowna down to Penticton is the nicest stretch of the trail. You can find more information here.
In Vancouver you are only a few hours from the West Coast Trail which is by far the best hiking trail in Canada. They allow for 10 unreserved hikers per day so you might be able to just show up and get in a hike.
The sunshine coast is an amazing place and would be a good place to go exploring. Similar coast line scenery as the WCT but much less harsh and better weather.
Wells Grey National Park is a little further of a drive but the Waterfalls are suppose to be amazing.
A little closer to home (Vancouver) there are a bunch of trails but probably the most well known one is the Grouse Grind, it will only take you an afternoon but it is one you will certainly remember. I see that the trail is currently closed but hopefully it will be open in time for you trip.
Have fun and I hope you enjoy your time here in the North.