Menton & Monaco; Eze and Cap Ferrat

We took a bus from Nice, very easy. We went to Cap Ferrat but were not interested in visiting Eze.

There are a couple of worthwhile hillside gardens in Menton in addition to the (typically very touristy, but attractive) historic center. One of the gardens is walkable from the town's second (easternmost) train station, bit some form of transportation will be necessary for the other one, I think. Buses go up there, but I don't know how frequently.

I haven't been to Monaco, so I'm not sure how practical it would be to take a Memton-Monaco day trip

Menton and Monaco are doable in a day. Just start early and head to Menton. It is not that big, unless you are very interested in the Jardins or a meal. Then hit Monaco on the way back, where the activities can stretch through dinner.

You can do Eze and Cap Ferrat by public transport. From Antibes, I'd take the train to the Eze station and then the bus to the top. You can also hike (or the sons can) up the steep path (proper footwear, not sandals). Another option is an uber or taxi to the top. You can also take an Uber from the village or taxi to the Cap Ferrat. Otherwise, a train from Eze to Beaulieu-sur-Mer. Just under the tracks is the bus stop to the Villa Ephrussi.

You might find it more efficient by car, and if you do, add a trip to La Turbie and or Villefranche-sur-Mer to your journey.