Roaming Riviera from Vieux Nice

Visiting Riviera towns from Nice was the easiest
vacation we have done in Europe! Lucky you, staying at La Perouse! We have been to Nice many times and love it with it’s mix of French and Italian cultures.
You take buses along the pretty waterfront except for Antibes where you need to take the train which is away from the water. We loved Antibes! You go to the central square in Nice to board the buses ( check location to be sure) , very inexpensive. Our favorite day trips were Antibes, Villefranche, St-Jean Cap Ferrat(a big favorite!) Beaulieu sul Mer, where you can visit the reproduction of an ancient Greek villa, St-Paul de Vence, and Eze.
The Villa Ephrussi in Cap Ferrat is a very worthwhile visit as is Villa Kerylos in Beaulieu, the reproduction of an ancient Greek villa.

Some coastal destinations are faster by train than by bus; the traffic along the Promenade des Anglais can be very slow--and unpredictably so, I was told.

I think the tram will work well for getting you between La Perouse and the train station as necessary. Google Maps indicates it's about a 9-minute walk from your hotel to the place where you'd board the tram, and later a 3-minute walk from the nearest tram stop to the train station. I'd imagine plenty of taxis will be happy for fares from that hotel as well.

If the Petit Palais you're referring to is the hotel on Av. Emile Bieckert, it looks as if that would be an 18-minute walk from the train station. It's unclear to me from Google Maps whether buses would be of assistance along that route. In addition, from looking at the squiggly nearby streets, I wonder whether the Petit Palais is up on a hill.

There are trams that run from the train station to/from old town locations - Place Massena is one. I stayed in old town Nice and used the tram to get to the train station. When I did a day trip to Vence and St Paul de Vence, I took the bus and it stopped a few blocks from my hotel. So day trips by train/bus were easy from my old town location. I'm not sure I would switch my hotels just for that reason.

Thank you all, happy to put my mind to rest on this one;.

Great that you are sticking with La Perouse, which was recently renovated and is close to restaurants, bars, cafes and the Promenade. The views are terrific as well. Le Petit Palais is actually up the hill in Cimiez -- not all the way up but with some elevation gain, further, and not the best/most interesting long walk to the station (past some grittier areas close to the elevated highway). You can take the tram from the Old Town. If you had wanted to stay walkable to the train station but still near restaurants and the water, you would look in the Carre D'Or, starting at Blvd Victor Hugo and south to the sea.

Sad that I didn’t book a room with a view but have been very excited to stay there.