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I'm not familiar with driving in India, and I'm not sure what you mean by a "new driver", so maybe the simplest thing to do is to give a brief description of the roads on the route that you describe. Then I'll add some comments and suggestions.
General comments: All the roads in Alberta are well-maintained. However, the warmer months (May to October) are the only times when road maintenance is practical (ground isn't frozen), so your family may see a certain amount of roadworks, such as repaving, filling potholes that were created in the spring, and just general road construction.
Avoid driving on the highways during twilight hours (sunrise/sunset), as wildlife like deer are more active then, and are hard to see when they cross the road. In some places in Canada, collisions with deer are the major cause of traffic fatalities, so don't take this risk lightly.
Calgary to Banff: The first 45-60 minutes will be driving through Calgary. If they are staying near the airport, I would recommend taking Stoney Trail (Hwy 201) to the west edge of the city, where it joins up with Hwy 1 (Trans-Canada Hwy). Stoney Trail is a freeway with a speed limit of 100km/h through most of this section. Although there are many exits, there are no traffic lights and there are no signs to let travellers know what services (food, fuel, shopping) might be available if they take any particular exit.
If they're not comfortable with driving at higher speeds right away, and want to be able to stop for food, fuel, etc., then Hwy 1 (Trans-Canada Hwy, 16th Avenue N within the city limits) could be a better choice. It's a slower road (60 km/h for most of its length within the city) but it makes the navigation easier and it's easier to make stops. From the west edge of the city, the speed limit on the Trans-Canada rises to 80 km/h (from just west of Home Road) all the way to the city limits. Don't speed, even if all the other drivers are doing it! Although you may see other drivers speeding past you, I can assure you that this section of the Trans-Canada is a favorite spot for photo-radar speed enforcement (I see them there on almost a daily basis) and the car rental companies *will* recoup the cost of photo radar tickets that they receive from the people renting the cars - plus handling fees.
Once out of the city limits, the speed limit on the Trans-Canada (Hwy 1) rises to 110 km/h all the way to the Banff park gates. This is a pretty route, over some gently rolling hills as you gradually climb into the mountains. There are no hairpin turns or sharp drop-offs on the sides of the road, and the road is wide and well-engineered for this speed.
Once inside the Banff park gates, about 60 minutes west of the city limits, the speed limit drops to 100, but the road continues to be excellent.
Banff to Jasper: The best route to take is the Bow Valley Parkway (Hwy 1A) to Lake Louise. This is a scenic road with a slower speed limit (60 km/h).
From Lake Louise, you will take the Icefields Parkway (Hwy 93N) to Jasper. This is an extremely scenic route with many places to pull over and admire the scenery or take short walks to scenic destinations like Peyto Lake and Sunwapta Falls. Although the posted speed limit is 90 km/h (if I recall correctly), people often drive more slowly for a host of reasons, ranging from enjoying the scenery to "bear jams". Again, not much to worry about.
As for snow, in the mountains there is a small chance of snowfall in any month of the year. But in the warmer months, all you need to do is wait for it to melt, which doesn't take very long.
DON'T drive to Edmonton and then Calgary, unless there are relatives, friends, or business associates there which must be visited. Those roads are extremely boring and long drives. Instead, return to Banff via the Icefields Parkway (enjoying seeing the scenery from a different point of view on the return) and then to Calgary from there.