The Alps

We need input from someone who has actually been there, but you can get from Nice to Grenoble in 5-1/2 to 7 hours by train, with one to three transfers required. You'd be swinging west through Avignon and Valence.

I understand from an acquaintance living in that part of France that Grenoble itself is not a particularly charming city but does have good museums. As for a picturesque smaller town accessible from Grenoble, there I cannot help you.

There might be a suitable location closer geographically. There are both national and regional parks south of Grenoble, around the town of Gap. But that area might not be any faster to get to, especially without a car.

To get to a true Alpine village, you may need longer than a day from Eze (by train). There is a train to Tende in the maritime Alpes (Train des Merveilles during the summer). There is also the the Chemin de Fer de Provence (single gauge). Both depart from Nice. These are much lower mountains than the true high Alps.

Grenoble is an OK city but not worth such a division to go (locals may argue, I apologize). It's in the valley, as you may know, so to get to elevation you need to get to one of the many lifts (I've not done it).

Which "alps" are you referring to? Getting to any of the alps is likely going to require almost a full travel day in each direction. For example if you wanted to see Mont Blanc in the French alps, a trip from Eze to Geneva is going to take ~8 hours with several changes, and then another hour or so from there to Chamonix.

You may already be aware of this, but note that your Rail pass does not include the compulsory reservations on the French TGV high speed trains. You'll have to pay out-of-pocket for the reservations, and reservations for pass holders are limited. Once they're sold out for a particular train, you wouldn't be able to use your passes on that train but would have to buy regular tickets.

I'm not sure March will be the best time for the alps, as that's kind of between ski season and summer tourist season and some tourist facilities may be closed.

No where easily or close via train.
Personally I would rent a car and drive 5-6 hours each way to Chamonix OR Fly from Nice to Geneva and catch a train there to Bernese Oberland area, Chamonix or Zermatt.

Taking the train there would not make sense ; there are likely some Italian towns in the alps that could be reached hours quicker by train since Eze/Nice is so close to Italy but still you are looking at an all day train ride each way.

We found Chamonix delightful and loved the Glacier experience. There's hiking and a gondola up to Mont Blanc. Lots of shops and restaurants, too.

However, it's not exactly the easiest city to get too. All those mountains make travel more complicated whether by train or car :)

There are some shuttles that run from Geneva to Chamonix. You might try the reverse route of what we did. Take the train from Nice to Geneva (or fly); then a shuttle to Chamonix.

Enjoy!