Keeley, it depends on how young your kids are. We are going on our 4th this June and have always gone with Princess. They have a very good kids program on board for ages 3 and up, all the way to teens. They have activities all day and you can even leave them there when you go on excursions. We went on our first cruise in June 2007 when our girls were 7. It was the Baltics and they enjoyed it. We didn't know any better and booked a lot of ship excursions, which you don't need to do in a lot of places if you do a little research ahead of time and don't mind getting out there on your own. Check CruiseCritic.com for passenger reviews and info on ports. We did a Med/Egypt later that year and took them out with us to the major ports. Is that the one you're looking at? You can definitely get around on your own very cheaply in Istanbul. Athens, maybe not so much and Cairo you definitely want to book an excursion with the ship or privately, because it's about 3 hours away from the port. We went through the ship but if I were going again, I'd do a private tour, because after sitting on the bus for 3 hours in a caravan of 168 cruise tour buses, they gave us about 45 minutes to walk around and ride a camel, another 30 minutes to see the sphinx and pyramids, and about an hour to shop in a store, probably owned by the bus driver's brother LOL. Lunch at the Cairo Hilton was included, but I felt like we didn't see very much of Cairo.
If your cruise begins in Civitavecchia rent a car here on base and drive up. Drop the luggage, mom and the kids at the ship and drop off the car about 7 blocks away. You can see the ship from the rental drop off. If it ends in Rome do the same thing, just reserve it here for a cheaper rate.