flights

If Venice works for you, then Venice Marco Polo airport, Code VCE. There are not that many direct flights from the US, so getting there (or leaving) usually means a connection in the US at a major hub or in Europe (Paris, London, Amsterdam, Frankfurt, depending on your carrier, maybe even Rome.).

For Rome, the main airport is Fiumicino Leonardo da Vinci Airport, code FCO.

If flights into Venice prove difficult or much more expensive, flying round trip to Rome is not bad, about a four hour train ride to Venice from Rome. You could also look at an airport like Florence (Code FLR), which would save you an hour or two by train, about the same as Milan (Code: MXP).

It all comes down to what price and schedule work for you.

You can buy an open jaw, multi-city ticket, so you aren't back tracking. We have found flying into Milan to be less expensive than Venice. However, it's quite scenic to fly into Venice. Avoid flying out of Venice, especially for morning flights. It's difficult getting to the Venice airport early in the morning.
So perhaps fly into Milan or Venice, and out of Rome.
Good luck!