I know Royal Caribbean offer special kids fares & I'm sure they're offering Med cruises this summer. Their ships are great for families too. Why not call cruise.co.uk for a quote.
There is a excellent cruise starting/finishing in Venice it covers Napels/Athens/Rome/Villefranche/Santorini and with most companies it is 10/12 days duration. If going into Venice with a balcony book a starboards stateroom if leaving then port as the view leaving/arriving are some of the best in the world with whole city unfolding in front of your very eyes as you slowly move up/down to/from the berthing point at the pier.
In most cases cruise lines will not allow under 18 to occupy cabins on their own. If we dont have a 4 berth its me and my son in one wife and daughter in the other.So if its 2 adults and 4 kids a 2 berth outside and 4 berth inside with the adults split on the booking.
Also I recently sailed with NCL and was sold a cabin as a 4 berth outside. What they didnt say was it was a small double sofa bed for my 16 year old daughter and 17 year old son to share! So make sure you clarify that.