Visiting the French Alps in summer

Dear All,

I'm looking for some advice. I am planning a summer trip which will start with a few days in Paris and ending with a few days on the coast in southern France. In between, we'd like to visit the French Alps. We will be travelling by public transport. I'd planned on the TGV to Lyon then either a bus or train to the mountains. I'm not a skier and have no real knowledge of which parts of the Alps are particularly spectacular to visit.

From my googling Chamonix would be spectcular, but is 4hours plus each way on the train from Lyon, including the steepest train in Europe (?). We only have a couple of days in that area so i'm not sure if this will be time worth spent or not.

My question for you, forum-goers, is this, where is easily accessible by public transport to get a good view of some very big mountains?
Has anyone gotten the train to Chamonix and can tell me about the journey?

Thanks in advance!

Nic