A city tour which brings you up to a viewpoint where you can see Nice's main airport runway which starts and ends in the sea I think and you will have an amazing view of the city. Elton John has a house there on a cliff overlooking the sea. Of course a walk/drive along the promenade is a must. The city has a beautiful park, pedestrianised areas & fountains worth a stroll.
Next you are very close to Monaco so it would be worth visiting it. The train will get you there in no time but bear in mind once you get near Monaco it goes underground so you won't see any wonderful coastal vistas or vistas of Monaco at that stage. You can take a hop on hop off trip there. The two main tourist areas are the old town around the palace, the other area is around the casino.
The other option is to take a tourist train into the Alps. The train des Mescailles or something. The earliest train in the morning, which leaves about 8am (I'm not sure if its every day) has a multi-lingual commentary for tourists. You can get off at the wonderful cliff hugging town of Saorge, which is an easy short say 10-15 walk uphill from the train station. You need to be fit to negotiate the town since there are steps everywhere. There is also a monastery and a cafe when you arrive at the village. Another popular stop is the town of Tende but I didn't get much further than the restaurant near the station. You can continue on into Italy where the train terminates at the town of Cuneo. On the way back you can eventually diverge on to another route to the Italian Riveria town of Ventimiglia and then catch a train along the scenic coast back to Nice.
The other scenic train trip from Nice to Dignes les Bains which is further west, you can do part of the line by steam train if you're there at the right time mid journey. Entrevaux is well worth getting off at. It is a walled citadel with drawbridge and a medieval town. People can use this train to connect with Grenoble in the French Alps however when I was using it the train only went to far and pax who wanted to connect with Dignes les Bains had to use a coach.
http://www.beyond.fr/travel/railpignes.html almost 8 years ago