The Med would be a good option it will be warm enough for the kids to run around and have fun with out being so hot that it will burn through any sunscreen. Pick a fun ship such as NCL or Royal Caribbean they both have loads to do for kids and you could loose them for the duration of the holiday. These ships also have a sitting service on board and kids clubs during the day.
Your main problem is going to be cabins. Assuming you are a family of 2 adults and 3 children. Most cabins accommodate 2 to 4. Some lines such as RCI do have family cabins or you could take 2 adjoining cabins but this can work out to be expensive. Another alternative is to get a couple more family members to join your party and book a three berth and a four berth which can work out to good value. Kids are very well looked after and will have a lot of fun, there is always plenty for them to do. pick a good line such as Royal Caribbean, P&O Carniva.