Rome to Milan

Don't fly. Its only 3 hours by high speed train. Flying will take you 5 hours when you account for time to Rome airport, check-in time before flight and security, and time getting off and getting into Milan.

Again, train is the best way to Lugano (assume that is where you want to go). Takes 1 1/4 hour on frequent direct trains.

https://www.trenitalia.com/en.html
Use "Roma Termini", "Milano Centrale", and "Lugano" as stations.

Antica Locanda Leonardo gets good reports and is next door to the church where the "Last Supper" is located.

I'd recommend you look at train service instead of flying. The train between Rome and Milan is 3 hours, which is much faster than flying when you take into consideration wait time, security and travel to the airports. Both Rome and Milan airports are outside the city center and take time and money to get into town. The central train stations are located near everything you may want to see. I believe Rome and Milan only have one airport.
The Last Supper and Duomo are in completely different areas of town, but easily accessible to each other by the subway system. The area around the Last Supper is a beautiful, leafy neighbourhood. The area around the Duomo is a busy, commercial hub. It really depends on what your looking for and what other things you will want to see.
Again, I'd take the train from Milan to Logano.

Leonardo da Vinci–Fiumicino Airport (FCO) -- main airport.

Milan Linate Airport (LIN) -- again, main airport.

Travel time: 1 hr 10 min

Flights are frequent-- like hourly. One way cost is under 100 typically.

As mentioned, there is a train option.

"Antica Locanda Leonardo" -- is one hotel right by it. Suggest you google Last Supper in maps and search hotels nearby. Many choices.

Getting Lugano is around an hour by taxi, two by train (EDIT: bus + train combo according to google maps from Hotel by last supper).

Happy travels!

The only flights to Milan are on ITA Airways from Rome FCO Airport. And they go into Milan's Linate Airport.

You'd catch a 1/2 hour bus ride into the Milan Train Station. Then it's about 1.5 hours to Lugano by train from there.

Getting Lugano is around an hour by taxi, two by train.

Sorry, it is 75 minutes by train. Cost is 28 EUR if you take the EC train, or about 17 EUR on the hourly RE 80. These all go from Milano Centrale. I'm guessing 200-250 EUR by taxi.

Thank you EVERYONE! Will take the train!