how can I get by bus from Portland Rose Garden to downtown Vancouver, Washington?

Take Max to the Delta Park/Expo Center/Vanport stop at the north end of the Yellow Line. http://www.trimet.org/schedules/maxyellowline.htm

Then catch a C-Tran bus route 44 or 47 and get off at the next stop, or route 4 and get off at the second stop. http://c-tran.com/system-maps/c-tran-full-system-m...

I'd use the MAX and then take a taxi, honestly. You can take the red/blue line (doesn't matter) to Pioneer Square, and then take the yellow line toward Expo Center. Go all the way to Expo and then a cab should cost you $12 to get to Vancouver's waterfront. http://www.taxiautofare.com/us/taxi-fare/Portland-...

This isn't ideal, but since you're going between two states, the public transportation gets awkward even though it's not that far of a distance. Your next best option (to my knowledge) is to take the MAX to Union Square and then take Amtrak up to Vancouver, and then deal with Vancouver's public transportation system to get downtown. This would likely take a while to pull off and I'd just spend the extra $10 for convenience.

PS: You may want to take this out of the Canada section, heh.

You have run into a common problem, by putting Vancouver in the question it is automatically posted in the Vancouver, BC, Canada section. You may want to move it to the Portland, OR section (preferably) or perhaps the Seattle, WA section. Tried to do if for you and it appears to have worked....fingers crossed.

Vancouver Washington is not in Canada.