Google Flights is my favorite place to check for flights and fares. You say you'll be in Italy May 12 to May 19. I looked at nonstop, roundtrip flights from NYC JFK to Milan Malpensa MXP. I chose departing JFK on 11 May so you would arrive on 12 May and departing on 20 May from MXP so that you would have a full week in Florence. You could take a fast train between Florence and Milan.
All RT nonstops listed were cheaper per person than the price you quoted. The cheapest when I looked was $783 on Emirates.
I highly recommend that you use Google Flights to search for options, set up price tracking if you want and book the flight you want directly with the airline when you see a good price. But $783 RT per person sounds pretty amazing to me.
It's about 1 hour 45 minutes by fast Frecciarossa train from Milano Centrale to Firenze S.M. Novella. It's too early to book for May yet, and because flights can be late, I checked prices on Trenitalia to go next week. Prices vary based on time of departure. The standard base price per person would be about 55€. There were many other lower prices depending on time of day, but even if you have to pay full price, that seems pretty cheap to me.
By booking ahead you could save as much as half on the Firenze to Milano trip, because you will have more control over your time heading back home. All Frecce trains require reserved seats. I love them. The fastest I went between Rome and Bologna in June was 250 km/h.