Delta is a SkyTeam partner with Alitalia (Italy's national airline), so they may give you better NS flight options. You could try an open jaw NYC to Rome or Milan, then take the opposite back to NYC. There will be NS options in both directions. Search award calendar with the following multi-city:

JFK - FCO (Rome)
MXP (Milan) - JFK

...or into MXP & out of FCO...your choice.

My experiences with the train system in Italy have been wonderful, and there is an easy option to include Florence in your route between Rome & Milan (Rome to Florence: ~1.5 hrs / Milan to Florence ~1.75 hrs). ...make sure you get tickets on the fast train though; either FrecciaRossa or FrecciaArgento (Red Arrow / Silver Arrow). If you get on an inter-city train, it will take longer.

Pisa would be a quick side trip from Florence, and could be done in less than a day. The train takes about an hour each way. If you get off at the 2nd station (apologies; I don't remember the station name), you can see the cathedral, and it's a very short hike. The tower is next to the cathedral.

Suggestion for Rome: Spend time out at night. If you're only going to be there for ~3 days, don't get too worn out looking at the ruins to enjoy the really beautiful side of Rome. The Spanish Steps and the Trevi Fountain are well within walking distance of each other, and are both spectacular after dark.