DD used to live in the area so we've been to a few places.
Yes, the old town of Nice is quite sweet. I happened to be there once on a Monday when the museums were closed and spent a fab morning mooching around a bric-a-brac market close to the promenade.
By contrast, we found Cannes quite lacking in personality and dull, but that might have been bad luck when we were there - pre-film festival so all sorts of things were shut off and scaffolding being erected etc.
Eze is fab, if over- touristified. You get the train along the coast to Eze and then either get a bus up the hill or walk up the Nietsche path (or is it the Goethe path?). The best view in Eze is available in the Jardin Exotique, from the top of the castle remains.
If you want a lovely little hill village that's less touristy, I can recommend St Agnes, Gorbio or Castellar above Menton. Roquebrune is terribly sweet, and has a fab view over to Monaco.
Can second the recommendation for Villa Ephrussi de Rothschild on Cap Ferrat. We actually got off the train at Beaulieu and walked round the entire peninsula before going to the villa, but you might not be feeling that active. It did take much of the day.
I'm actually a big fan of Monaco because it's just so massively weird. You could easily spend a day there - go to the castle hill and visit the aquarium and the old town, go to the fab jardin exotique and cave complex (up the hill but there is a bus service), and then the Monte Carlo utter decadence. You can also play 'find the most tasteless yacht', which to be fair can be played at most places along that coast. I'm quite fond of Monaco in a jaw-dropping 'it's bonkers here' kind of way. DD and her friends used to go there for nights out because as a group of young women they would get free entry to the bars and even free drinks (with no expectation of anything in return).
Only for architecture fans: tours of the Eileen Grey / Le Corbusier complex below Roquebrune. A total highlight for me, but admittedly somewhat specialized.
I'm trying to think of the name of the little bay along the coast where DD and her friends always went swimming. Might well have been Baie de Saint Laurent, getting off the train at Cap d'Ail. Sandy beach, protected from the wind, next to a bar but completely free of charge to use.