Nice's bus and tram network is quite extensive, and so easy to use (but then again, I speak French, so of course I would find it easy).
And so cheap. Apart from the airport bus, for which you will pay €4, all bus and tram trips carry a fare of €1. Actually, the €4 fare for the airport is a day ticket, so keep your ticket if you're arriving early. But from what I could gather, the day ticket does not apply to Monaco. If you go to the Lignes d'Azur office on Place Masséna, they will give you a map of the area covered by the day ticket..
So we made good use of those buses - we went to St Paul de Vence (Bus number 400, about 45 minutes, every 45 minutes it would seem) and to Menton (via Monaco) - bus 100 leaves Nice every 15 minutes or so - now that journey took 2 hours, due to road works around Cap d'Ail, but once we got a seat, we didn't mind - the view was wonderful, as the road follows the coast for most of the way.
As it turns out, both buses currently leave from the same area, on opposite sides of the road, at the JC Bermond stop, which is a small road at the town end of this strange building that looks like an abandoned car park with plants hanging from the higher levels (the Google Satellite view shows a racing track - go figure?)
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Be prepared to fight your way onto the bus - the 100 bus was very busy, 2 buses were filled in no time at all. The French don't understand the concept of queuing, and I think they resent all those tourists swarming onto their beloved public transport. We did end up standing for a good half hour on the way to Menton - not very pleasant, but we survived, just glad it wasn't mid-summer. And some of the drivers seem to ignore all the rules - talking on their mobile phones while driving and chatting to the passengers standing near the entrance. But the roads were good, and our journeys were by and large smooth.