I'd look at RCI Navigator of the Seas - seems to fit your bill exactly! Either 10 or 12n out of Rome in Sept.
If you click on the cruise offers section and select the med and september you will see that there is quite a variety to pick from. have a look at which cruise has the ports of call that most appeal and then have a look at the description of the ship and cruise line. Different ships appeal to different people. Some like lots of formality, some don't. Some like lots of nationalities and some prefer their fellow passengers to be mainly British. then there is the question to go for one of the huge resort style ships or something smaller. Once you've narrowed your search down a bit, I'm sure people on this board will be able to give you their opinion on the specific cruises or ships you are considering. Good luck finding something. cruising is a fantastic holiday and I'm sure whatever you choose you will all have a fantastic tim.