Welcome!
I would certainly try to do Banff and Jasper - the highway between the two is one of the most spectacular drives in North America, if not the world. Plus Jasper has a different (more relaxed?) vibe than Banff. It's usually suggested to do 3-4 nights in Banff and 2-3 nights in Jasper. That said, if you only have 5 nights, perhaps split your time between Lake Louise/Banff/Canmore and do a day trip/tour to the Icefields Parkway.
It's generally far easier to explore the Rockies with your own car, but it can be done with a combination of buses and shuttles. In Banff there is local public transport (Roam) that will get you around the townsite and to Canmore. To get to Lake Louise/Moraine Lake (a must do day trip), you'd need to take one of the Brewster's shuttles/airport shuttles or arrange a tour. Greyhound also goes there. Note that most shuttles and the Greyhound bus go to the "mall" which is about 4km from the lake. From there you have to walk or take a taxi to the lake.
Between Lake Louise and Jasper, the only option on the Icefields Parkway is via tours or the Brewster's/Sundog shuttle. There is no public transport in the parks and the Greyhound buses go outside the parks via a VERY long and boring route (Banff to Calgary to Edmonton to Jasper). Another option are the Moosedog tours which are aimed at a 20-30 something crowd and use smaller buses and hostels.
In Jasper, there is no pubic transport. It can be a challenge getting to trailheads that aren't near the townsite. If you are into hiking, you may need to make friends at the hostels or do tours. For the closer trailheads, renting a bike might make things more accessible.
For accommodation, there are some great Hi Hostels in the area. In Banff, the Hi Hostel is large and gets great reviews - has a cafe, private rooms and is a bit from the centre of the townsite, but still easily accessible. In Lake Louise, the Hi Hostel is co run by the Alpine Club of Canada and is one of my favorites. Two large buildings each with a kitchen, nice common areas, cafe, free or low cost bike rentals, 4-6 person dorm rooms and private rooms. It is about 4km from the lake, but you can bike or walk to the lake. There's also an nice, but small Hi Hostel in Canmore, though it is a bit out of town and the Hostel Bear is good.
Jasper is a bit lacking in hostels - none of the hostels are within walking distance of the townsite. Whistlers is about 10km from town, and while it has modern conveniences (not a wilderness hostel) other than a few private rooms, it just has very large (20-30 person) dorm rooms that tend to be noisy. And the washrooms are not in great condition and are few for the number of persons in the hostel. The other hostels are also outside of town and have smaller dorms, but are wilderness hostels - i.e. no running water (other than tank water for cooking), outhouses and no showers. There is one great small private hostel in town (World Traveller's Hostel) that has a private room and one room with three very comfy bunkbeds. Great facilities and very personable hosts. Book ASAP as it fills very quickly.