Hi Tom,
June is a great time to visit BC as birdsong and breeding diversity is at it's peak. As migration has tailed off by early June, most banding stations are inactive during this period but the Colony Farm crew (which Larry linked to) often have something going on and are welcoming to visitors.
And as Jeremy said, a coast to Okanagan trip is the classic route and well worth it. You may also want to consider a couple days around Victoria (on Vancouver Island). The ferry from Tsawwassen to Swartz Bay (essentially Vancouver to Victoria) is a pleasant ride at that time of year, and you may be lucky enough to spot some Orcas.
As far as locations around Vancouver to visit. In June you really want to concentrate on the breeders, as shorebirds and most waterfowl have cleared out. These would be 4 of my top for a visiting birder to Vancouver at that time:
-Cypress Bowl Provincial Park (Forest and mountain species)
-Stanley Park (A great introduction to Pacific NW species (forest and on the water), and provides great views of downtown Vancouver and Burrard Inlet. Pigeon Guillemot and Pelagic Cormorant can be seen regularly at the north end of the park beneath the Lions Gate Bridge.
-Campbell Valley Regional Park (Wonderful lowland mixed forest and meadows); great way to see the declining Douglas Squirrel among many western bird specialties)
-Reifel Refuge (as mentioned previously--this reserve is probably at it's best outside of summer, however for the visitor, there will be plenty here to enjoy in June as well).
As for the Okanagan, the southern part of the valley is where it's at. Check out the guide-book for details but I would highly recommend Vaseux Lake, White Lake, Rd 22, and both the Shuttleworth & McKinney forestry roads as places to visit.
Hope you have a great trip!
Russ Cannings
Revelstoke, BC
russellcannings@shaw.ca