Menton and Antibes are easily accessible from Nice by train. Aside from its picturesque medieval center (which usually is packed with tourists during busy months), Menton town has at least two good gardens, both on hillsides. One is reasonably walkable from the town's second (easternmost) train station; the other can be reached by bus or taxi. Antibes has a good Picasso Museum as well as an atmospheric historic district.
For St-Paul-de-Vence you'd take a bus from Nice. I like the Galerie Maeght (modern art) just outside the village. It's walkable from the nearest bus stop. St-Paul-de-Vence tends to be mobbed during day-tripping hours, unfortunately. It's not large, and I was ready to leave very soon but was very happy to have had the opportunity to see Galerie Maeght. Beyond St-Paul-de-Vence, and reachable on the same bus, is the larger and comparatively ignored town of Vence. I liked it much more. However, if you're there during the after-lunch siesta period, you may find it not exactly lively. I've been told Tourrettes-sur-Loup, which is further beyond Vence, is mostly ignored by tourists and very nice.
Nice itself has enough sights to fill more than a day, starting with at least six art museums. Other possibilities in the area are Eze Village (bus from Nice) and Villefranche-sur-Mer (train) to the east, which could probably be seen on the same day. To the west of Nice you have Biot (Leger Museum and glass museum; train then bus from Nice), Cannes (train) and Grasse (way up in the hills but accessible by train). Of those last three I've only been to Biot; I've read that Grasse is nothing special unless you have a real interest in perfume.
One other possibility I enjoyed was the trip on the tourist-oriented Tren des Pignes, which runs from a dedicated train station in Nice (not Nice Ville) to medieval Entrevaux and the spa town of Digne-les-Bains. I took that trip on an extraordinarily rainy day when no one else cared to be out and about, so I cannot comment on how tourist-filled Entrevaux and Digne would typically be. The train doesn't run often, so it's not particularly easy to schedule the trip so that you have time to walk around both Entrevaux and Digne. I think it's safe to say most visitors would be more interested in extremely atmospheric Entrevaux, but that may mean it's usually overrun. When you get to Digne there's a bit of a walk from the train station to the village.
A final possibility would be to take the train east beyond Menton to one of the Italian coastal towns like San Remo. I haven't done that.