You can rent same car from San Diego to Bellingham, WA (between Seattle and Vancouver) for about $175/day.
Same car to Jackson, WY or Whitefish, MT (near Glacier NP) is a horrible $275/day.
You can get to any of the above locations in 2 or 3 days of driving. Figure $4.50/gallon gas at 22mpg to play it safe.
A fare and # of transfers by bus (Greyhound) to SLC $90/2 Jackson $100/4, Whitefish $109/5, Bellingham $109/4, and Jasper $119/5. Bike fees (maybe $20 each) are extra. If traveling out of Los Angeles,transfer tranfer required.
Airfare to SLC is $109*; Spokane $248*; Jackson $500; Missoula, MT $225; Edmonton$180; Calgary $160 ($190 NS). Bags and bikes are extra $20-$50 each). * = first 2 bags free.
You could do a combo of things. For instance, say you want to see the Grand Canyon on your way north to Jackson. You could rent a car for two days to SLC. Then take a no-transfer bus (Mountain States Express (MSE) NOT Greyhound) for $70 (up to 3 pieces of "luggage or bikes) to Jackson. This would cost about $300 for car and $70 per person for the bus. If you wanted least amount of transfers would be to take a bus from Los Angeles to SLC then trantotalingSE totalling 3 transfers.
Finally, if you could start in latter part of June, I would think that you could start in Edmonton and work your way south without danger of snow.