I agree with your friends. Florence and Venice do have airports, but often they are not the best choice.
The Florence airport is small, and there are fewer carriers serving it, which means less price competition, and often less flexibility in schedules. When my son lived in Florence for a year, he never used that airport; he went to Pisa instead for infra-Europe flights, and flew home from Milan.
I enjoy flying out of the Venice airport for the views, but only for an intra-Europe flight, never a flight home. Most of the airlines have you connect somewhere in Europe for a flight to the USA, and that mens a very early departure from Venice. I have lost track of how many posts we have seen on how to get tomthe Venice airport early in the morning. For some flights the only answer is an expensive water taxi, because the bus and vaporetto are not yet running.
The exception would be if there is a direct flight from Venice to the US, but there are only a few cities served by such flights, and often they are seasonal (summer only).
Milan is a good choice; it is a short and cheap train ride from either Venice or Florence. In addition to the Last Supper, one can see the magnificent Duomo. I have always wanted to go up on the roof, but never seem to make time for that. Someday . . .