You can rent a Toyota RAV4 from San Diego to Salt Lake City for about $150/day (based on weekend). Ride from SLC to Jackson, WY and see Tetons and Yellowstone before heading to Canada.
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.