My best recommendation will be the bus.
Your main concern is the fact that bus itineraries are not flexible enough. If you take the car to Whistler, then want to drive to the Rockies, your smoothest and fastest itinerary will take you through Vancouver too.
A car is costly, and if you drive it one way you will have to drive it back. Last summer I did that (with an even more customized trip) and ended up driving the car back from Calgary to Vancouver with someone to share the ride on part of the way, it was a long 20-hour drive (with a sleep break).
I posted my ride share ad on kangaride.ca and found only a French guy to share the ride (he had been to Eastern Canada, where kangaride/amigo express is popular). I posted it on hitchplanet.com (which is a ride share website more for Western Canada) too but got no one interested, nor on Craigslist. I didn't try but kijiji.ca is popular in Canada too. In Eastern Canada there are also some local facebook groups where people share their trips. And a final way to find ride shares is to post ads in hostels (email them ahead of time, it might help). Here are your ride share website options.
I would not count too much on them though, there are definitely fewer options than in Eastern Canada (and way way fewer than in Europe). My experience in Eastern Canada vs France is that Eastern Canada has a many rides per day between big cities and about one per day to remote areas; in France it is multiple per hour between big cities and multiple per day to remote areas. Western Canada seems to have one every couple days, even between Vancouver and Calgary. If you plan your trip early enough, you might want to post on travel websites for finding travel partners (I can think of pvtistes.net, it is a website for French people going on work and holiday visas, mostly to Eastern Canada and there are a lot of ads for shared trips).
For the bus, you definitely have the perfect itinerary, you follow exactly the bus network. Vancouver-Whistler seems to definitely be a touristy route (sure you will come back to Vancouver to go East but by car probably too). The "sea-to-sky" highway (the road between Vancouver and Whistler) is definitely one of the most gorgeous route I have travelled. I drove it and did not enjoy it enough (winding a lot, I had to focus on the road) and there are very few parkings to stop on the way. So I would recommend to take the bus on that portion and sit on the left of the bus on your way up to Whistler. The good thing is that the price for this route is fixed so you can book it just one day ahead if you don't find a ride share.
Then the Greyhound goes all the way from Vancouver to Calgary through Banff and Lake Louise. Lake Louise bus stop is several kilometres from the lake itself but definitely walkable in the morning to reach your trailhead. And Banff might be a little better with a car since it is a large town and trailheads are a bit out of town. Greyhound offers cheap advance fare (more than 21 days before) so these would be the tickets to book early on.
The part where a car might be the best option is along the Ice fields parkway, linking Banff to Jasper. This is a national park and it offers gorgeous views from the side of the road or from the top of some 1-hour hikes from the road. You can still see many of these from the bus and actually hike from Jasper. There are many trailheads from Jasper town (but a much larger choice if you have a car there). Brewster (the bus company) also offers Ice fields parkway trips but they seem a bit overpriced.
Hitch-hiking (not on highways) from Banff or Jasper to a trailhead or back is certainly easy.
Oh and I almost forgot the train. The best views from the train are East of Jasper (not on your route). And the westbound train from Jasper to Vancouver rides at night so you don't see much anyway. If you do not want to spend too much money that's not the best idea. The views between Lake Louise and Yoho park are more worth too.
So here are all your options: the car for the flexibility but for a cost and you have to drive it back (or pay an expensive one-way fee), or a mix of bus and ride shares (most likely just bus) for cheaper but with much less flexibility. Your turn to choose.
On top of that, if you pick the car option, I would definitely recommend you to go a bit off the beaten paths, i.e. not stay in downtown hostels but the so-called "wilderness" ones, going to other parks on your way (Garibaldi lake near Whistler, Kootenay, Yoho, ...) and obviously offer to share your ride on websites.
Note for the bus: on many buses, checks are performed prior to boarding and bear spray is forbidden (even in checked bag). So if you need bear spray, buy it locally.