Here are few simple options - within about an hour of Vancouver. Others just a few minutes away from downtown.
Jig off the pier at Ambleside for flounder, sharks etc, go up to Rice Lake and drop a line through the ice for rainbows. When ice is off, you can rent a canoe or kayak and paddle around Burnaby Lake dragging a leach or wooly bugger pattern. You could try your hand at steelheading on the Vedder or on the Sea to Sky, find a guiding trip/intro lesson on steelheading if you are new to it.
Like others have said, there are a million options including the local ocean for salmon. Come August, you can find some shore anywhere along the Fraser River (follow the parked cars) and cast for Pink Salmon. You'll have a blast with that i'm sure.
You might want to get a feel for the type of fishing you'd like to try, because much of it requires specific/dedicated equipment and tackle. If you give us a sense of the the fishing you have done in the past and what gear you currently have, we can provide options that can get you started without a lot of outlay. Then you can go from there.
As noted above, FishingwithRod has some videos to watch on fishing options (many of which are in the lowermainland and accessible) for beginners and beyond;
http://www.fishingwithrod.com/video/
There are years of videos here beginning in 2010+