If it was me, I would spend a couple nights or so in Seattle. It is a good city to visit. We stay right downtown at the Grand Hyatt. I would rent the car in Seattle and drive up to British Columbia. You should have no problem taking the rental car into Canada. I have taken a rental car from Washington into British Columbia with no problems. You either drive straight from Seattle to Salmon Arm bypassing Vancouver or you can drive up to Vancouver if you want to see the city.
Renting the car in Seattle will save you a lot of money and hassle. Coordinating with the train is a pain and you have a lot more flexibility if you just rent the car when you arrive in Seattle and drop it off when you leave.
I would go with John's reccomendation. I frequently drive to Seattle and fly from there so much cheaper than Vancouver where I live. Coincidentally, I am planning a Seattle to New Orleans flight later this year. Southwest Air is just so much cheaper than anything direct from Vancouver.
June will be a great time of year, weather should be good, but little chance of being stifling hot and the usual fire bans. And of course the crowds are less.
From Vancouver to Shuswap you have a great choice of circle routes to drive. The fastest route is approx 4 hours from Vancouver. Sea-Tac to Vancouver is 3 hours drive with an average border wait. (Try not to cross Sunday afternoon/evening).
Examples of the routes you can take:
Outbound: Vancouver to Kamloops/Shuswap via Fraser Canyon (#1) or Coquihalla (#5)
Return: (a) Fraser Canyon or Coquihalla whichever you didn't take.
(b) Okanagan Valley wine country and then Hope-Princeton (#3) to Vancouver
(c) Okanagan Valley through to Washington State then back to Seattle via Stevens Pass/Leavenworth
(d) From Kamloops to Lilloet, then on the Duffy Lake Road to Whistler, then the Sea to Sky back to Vancouver
Give me an idea of the time you have and your interests and I could suggest stops along the way, depending on your needs. For example, using the Whistler route you could do a 1 hour hike to a glacier lake (could even fish there), ride the peak to peak the highest gondola in the world, zipline, etc..
If you take the Hope-Princeton route, you can visit wineries, and I could suggest a day hike into the alpine along the way, the wild flowers should be blooming by late June.
One last thing, you need a fresh water fishing license (not salt water) here in BC here is the link, you can make an e purchase.
http://www.fishing.gov.bc.ca/
Have a great trip
Randy