No one has mentioned Genoa yet, IMHO is a very, very underrated city with a really huge medieval old town, great restaurants and bars and as well as the old city there are some fabulous places along the coast 20-30km either side, all served by frequent coastal trains. I think the reason it is not well known is probably because when people think of Italy for a break they generally think of Rome, Venice, Florence, Naples etc.
If you head round the coast by train from Genoa (in the Rome direction) there is Nervi, Camogli, Santa Margherita, Portofino, Zoagli, Sestri Levante - all beautiful places with stunning beaches and great restaurants. Genoa airport (GOA) is only 1.3km from a main coastal line train station....so either a 15 minute level walk away or there is a frequent 5 minute shuttle bus. Genoa city center is just 9 minutes once on the train.
As others have commented I can also recommend Turin too, lovely huge old squares and great bars and restaurants but would still prefer Genoa because of the coastal location which offers more options to combine a city break with a beach location too.
It must be being discovered more and more now as Easyjet started flying there in 2018 from LTN, BRS and AMS
