After Nice, France suggestions
 

Moved to Europe

You have a wealth of options, as the public transportation network in that part of France is excellent. You can explore your options with a good guidebook or two, or using rome2rio.com, or by reading trip reports on this forum. If you say something about your interests, Fodorites would be better positioned to offer suggestions.

Enjoy!

I have been to Nice myself, and if I had a chance to stay longer I would have visited Monaco or not too far from Nice - Marseille. Since you haven't provided any additional info on your travel interests, I'd suggest to look up these places firstly as they are super convenient for traveling from Nice.

Why not go over to Italy for a few nights - we loved Rapallo - easy to get there by train. But where are you departing from? Not sure how that works into your trip plan. On one trip we went over to Aix - different vibe than Nice.

I wouldn't say Marseille is "not too far from Nice". It's about 200 km. Too far for a day trip by my terms. The question is too vague, it isn't clear if the OP is just looking for day trips or wants to change hotels and stay some place else. And how that figures into the comment about staying in Nice for a week, which sounds like some plan for a reason, maybe work or something. There are plenty of day trips by train like Antibes, etc. If you want to move hotels for some days, yeah, then I think Marseille would work, if that's the question. It's about a 2.5-3 hr train ride. It's about 3 hrs by TGV to Aix-en-Provence, also, another choice.

I didn't like Monaco, my wife did but then she's impressed by ostentatious displays of wealth whereas I find it a bit nauseating. We popped over to Sanremo in Italy (we drove but I believe it can be reached by train), a pleasant seaside town with better food than Nice (in my opinion). The beaches looked as uninviting as they did in Nice with effluent pipes leading not vey far out to sea but I wasn't there to frolic on the beach as we had a villa with a nice clean pool to return to. Sanremo is a bit more rough around the edge than Nice but it's Italy, it has it's own Italian charm that is lacking in Nice and certainly lacking in Monaco. I found it a nice change although comparable to the smaller villages surrounding Nice that didn't have the same air of arrogance that was slightly detectable in Nice.

Are you looking to switch up your base and still remain in the Cote d'Azur? If you are limited to public transport, most places along the Cote d'Azur coast are a simple day trip, although many transport links connect via Nice. If you wanted a smaller city, you could try Menton, as it is more Italianate. If you wanted a beach resort you could try Juan les Pins, but architecturally less interesting and is so close in anycase. .

If you are looking to head way out of the region, Avignon or Genoa are other locations to base. Or you could head to the Italian resort towns as mentioned above, but I prefer Boghidera to San Remo. If staying in France, you could head to Cassis along the coast near Marseilles.

Since you will be in Nice for a week, then take one of the longer trips and spend a couple of days based there. The other stuff you'll be able to do while staying with your Aunt. Consider San Remo/Borghidera or Genoa although they get their share of tourists. There are even more touristy places like Casiss or Avignon (with day trips via tours or transit). You might enjoy Montpellier as well, it has a very youthful vibe due to the Uni population

do you have to fly home from Nice - if not take the TGV to some place like Avignon - wonderful base for so many neat things and one of the most beautiful cities anywhere IMO. Fly home from nearby Lyon. For lots on trains check www.oui.sncf - French Railways where you can book your own TGV tickets at a discount if early enough - www.seat61.com tells lots about that - general info trains and where to go with them www.ricksteves.com and BETS-European Rail Experts.