$584 - Flights to Nice, France from New York
I found this flight price on a bunch of different dates, though you will usually need to fly on Tuesdays and Wednesdays. Flying on the weekends will cost a couple hundred dollars typically.
For an initial flight search, try flying out on September 29 (Tuesday) and flying back on October 8 (Wednesday)
Flights with TAP Airlines have a 2-4 layover in Lisbon, and take about 14 hours each way total.
If you have the time off in early October, I’d highly suggest getting on one of these flights. It won’t quite be swimming weather in Fall, but it will be plenty hot, and it will be the tourist off-season, so you can find cheaper hotels and hostels, plus have no problems getting a good seat at any of the most popular restaurants.
Nice tends to be a popular coastal destination for many Europeans (especially British) , so expect to encounter all types of travelers on your visit.
Here’s a nice summary of attractions in Nice as well.