The Sea to Sky Gondola is pretty cool and is about an hour north of Vancouver and is pretty cool. There are hiking opportunities and amazing views from the top. We did that in the morning and then went on to Whistler in the afternoon. The drive there is really amazing on it's own. Just be aware that traffic can get pretty bad on weekends. The Lost Souls of Gastown walking tour was pretty cool. Also, some pretty good bars in Gastown as well. The only thing to really avoid is the homeless encampment which is a little less glamorous side of Vancouver.
It's been many many years since I was in Vancouver on business related manners.
I would echo a couple of comments. Definitely check out Stanley Park. Named after Lord Stanley (same person whose name graces the Stanley Cup). The cedar trees there, and their diameter, just blew me away.
I never got to Vancouver Island, but have been told if you ever wanted to experience Great Britain in North America, go there.
Stanley Park, Capilano bridge and Granville Island are all cool and you can knock out at least 2 in one day if you plan it right. Vancouver is a really fun city, lots of cool areas to explore
June in Vancouver is busy. Traffic is Vancouver is busy. Parking in Vancouver is busy. There are many bike rental shops in Vancouver. Rent a bike and tour the city. Therefore, you can easily stop and park to eat, shop, & see. Vancouver is not a big city. You can easily bike tour the entire city from Granville Island to Stanley Park to China Town on a bike.
All of the above suggestions are great. I think Vancouver may be my favorite city in North America. If you have time take the ferry to Victoria on Vancouver Island and visit Butchart Gardens.