I was thinking of flying into either Milan or Rome and then flying out
home from a different country. Is this a good idea?
No. Leaving NYC on Oct 12 and Italy on Oct 22 only gives you 9 nights/8.5 days to work with: barely enough time for 3 locations in Italy.
Wanted to visit a part of Tuscany. Any suggestions for the best
places?
If you want to visit Tuscany, then fly into Venice, skip Milan, and fly out of Rome: go to Florence (capital of Tuscany). Take a day trip from there by train or bus to Siena, or train to Lucca. Or skip Venice for Milan. In any case, you really don't have time for a 4th destination.
I have read that the best wat to get around Italy is via train. Is it
not recommended to rent a car?
For a plan that includes urban areas, trains are absolutely the best way to get around. A car is essentially worthless in Rome, Florence, Venice and Milan due to many pedestrianized areas (Venice is ENTIRELY pedestrianized) and vehicular streets which are restricted to locals (ZTLS). Parking is also a challenge. Trains are the fastest way to get from one city to the next when time is tight, as yours is. You can hop one in the middle of one city and be dumped out right in the middle of the next in shortest order.
All of these cities have good public transit and, frankly, some are really best explored on foot anyway. A car is a bonus if planning to spend a lot of time in the countryside but not for a plan which is focused on cities.
Anyway, an itinerary which skips Milan and includes Tuscany might look like this:
10/12: fly out of NYC
10/13: arrive Venice (3 nights) Flying INTO Venice is recommended as flights OUT are often really early in the morning
10/14: Venice
10/15: Venice
10/16: early train to Florence, Florence, (2 nights)
10/17: Day trip from Florence
10/18: Train to Rome, Rome (4 nights)
10/19: Rome
10/20: Rome
10/21: Rome
10:22: fly home from Rome
If there's ANY way to add another day or two I'd advise it so that you had 1 more for Florence. Or subtract one from Rome, although I consider 4 nights the minimum for that one as there's so much to see.
A plan which skips Venice might look like this:
10/12: fly out of NYC
10/13: arrive Rome (4 nights)
10/14: Rome
10/15: Rome
10/16: Rome
10/17: Train to Florence (4 nights)
10/18: Day trip from Florence
10/19: Florence
10/20: Day trip from Florence
10/21: Early train to Milan: you can sightsee enough highlights with a partial day.
10/22: Fly home from Milan Malpsensa (1 hour from central Milan)