My wife and I were out in Nice early May this year (and previously). Caught the bus from Nice bus station (near the old town) to Menton. It took quite a while and passed through Monte Carlo/Monaco so we had a scenic trip. We decided to travel back on the train for speed and although it was faster the market had been open in a town just over the Italian border and the train was packed....I mean packed.
We have previously used the bus to travel from St. Raphael to St. Tropez but again there's a train service.
I would be happy to use the train out of Nice both eastwards and westwards but would try to pick less busy times to travel and avoid market day over the Italian border.
Car driving can be frought. Done it a couple of times and have to say the driving skills are suspect. Most cars are dented and parking can be difficult. The coastal strip between Nice and Frejus is very busy but if you head inland the roads are less congested. Public transport (bus) to places such as Grass, Vence, Eze is good but I don't think there's any main railway routes. Mougins might be worth a drive to (10-15 miles north of Cannes) but I'd avoid the car in Cannes itself. Villefranche, just eastwards of Nice is walkable from the old Town part of Nice in about 2 hours...perfect for working up an appetite for a meal in one of the many fish restaurants at the old harbour. Watch for Mondays though, some close that day and lunchtime serving often ceases by 14:30.
Port Grimaud is unusual; be careful though as the train may pass through Grimaud and it's about 4-5 miles inland of Port Grimaud.
Been to Nice many times and know a little about the town. PM me if you have specific questions.
Cheers, Mike