In addition to being fairly far west, St.-Tropez doesn't have a train station, so you have to switch to a bus for the last part of the trip, which definitely adds to the time required.
To reach places off the coast (like Eze Village), you'd need a bus ride. You can take a bus all the way to Eze from Nice or you can take the train to Eze-sur-Mer and pick up a bus up to the village there (I'm guessing it's a different bus but am not sure). If you all have good knees, walking down from Eze to the coast might be nice; I haven't done that myself.
Villefranche-sur-Mer and Eze are both rather small, but Nice will fill the rest of your time. If you want to explore more, the train running along the coast (except to St-Tropez!) makes lots of places accessible. I think there's something to enjoy at just about every train stop along that coast; different travelers have different preferences. Other than the gardens, nothing about Monaco sounds all that interesting to me, so I haven't been there. Quite a few other folks here have been and were not impressed; others recommend it. Let's just say the comments about Monaco are more mixed than they are about many other places.
Rick has a guidebook focusing on Provence and the Riviera. Even though your stay is very short, with five people traveling, I think it's worthwhile to buy that book. We can't tell you what is in every town within reach of Nice. If you travel without a guidebook, it's easy to miss out on the place or experience you would like the most.
Do be careful with your personal possessions (especially purses/wallets/backpacks) on the train; tickets are cheap, so it is popular with pickpockets.