I’d recommend the Norwegian Epic. It’s does a 7 day cruise round the Med, and you can start the cruise in either Rome or Barcelona. I’d think you’d be better starting in Rome if you want to do some sightseeing there, as the port is about an hour by train from Rome, making a day trip from the ship a bit of a whirlwind!
The cruise also calls at Livorno (for Pisa/Florence) , Naples, Barcelona, Palma (Majorca) and Cannes.
It’s a big ship with loads to do, good restaurants and buffets included, or you can pay a little extra for the speciality restaurants which are lovely. And not at all formal, very relaxed feel, although you can dress up for dinner if you want.
The price (if your in the U.K.) also includes the service charges, and also the drinks package, which is fabulous if you like a drink on holiday - drinks prices on a cruise tend to be quite pricey, so it’s good to have drinks included.
We’ve been on the Epic twice, and had a great time