Nice to Monaco: Shall we take a bus or the train?
Nice to Provence: Want to do a self guided day trip to St Paul DeVence, Tourettes and Cannes. Are there buses we can take?
Q1. Depends on you. Bus cheaper, slower, more like a bus, delivers you to the door of the TI. Train faster, better views, less convenient at Nice end, at least as convenient if you know which of the nearly half dozen exits to take to your destination. Q2. Yes, trains and buses to Cannes (although why you'd want to when Antibes is much closer and 100 times better), buses to St Paul de Vence. Dunno about Tourettes - a quick peek at the bus website will give the answer.
Have you considered Cinque Terre??
Bus #510a runs between Vence and Tourrettes-sur-Loup. See the timetables here.
Peti, For the trip from Nice to Monaco, I'd suggest taking the Bus. The Bus will drop you at the base of the hill in Monaco, and from there you can either walk up the hill or take one of the local Buses for another €1. If you want to see the Changing of the Guard at the Palace, that takes place at 11:55 daily. Be sure to get there early to get a good vantage point, as the square starts to fill quite early. If you want to visit Monte Carlo, it's just across the Bay and a fairly easy walk. However, you can also take a local Bus. Note that a dress code is enforced in the Casino. I don't have any information on the other routes. Happy travels!