It's really down to whatever will work for you and how your dc will be with shore adventures - we did the Med Cruise a couple of years ago at the end of August with DS who was 8 at the time. Here are some things that might be worth thinking about though (not sure what itinerey the cruise you're on has but am assuming various stops in South of France and then down the west coast of Italy).
Firstly it will be very hot, therefore full day excursions might not be huge amounts of fun. We did Rome as that was what DS was dead keen to see, DH and I know it well enough to do our own thing there and we took it easy - hit the main sites (Colosseum and Forum) first, then a long lunch and took it easy in the afternoon with lots of stops for drinks, ice-cream etc. It was still very very hot though and we were grateful for the aircon coach. We also did a half day trip to Pompeii which worked really well as we were back on the ship in the afternoon relaxing.
If you want to I think you can in fact leave the kids in the Oceaneer's Club/Lab all day so you could go off and do excursions without them - if they are the sort of kids who like that kind of thing then that is an option perhaps.
On all the port days you can 'tender' or disembark onto shore at any point after the excursions have all set off so you could just do your own thing eg decide late morning that you want to go into Monte Carlo for a bit of a wander/lunch etc. There are loads of resources on line about doing your own thing in the ports, some people organise their own excursions and there is a whole industry around pre-booking outside of DCL if you want to.
A huge plus of staying on the ship on port days is that there are far fewer people on board, therefore the lunch buffet is less busy if you go there and the pools are so much less crowded (they are packed in the daytime, or at least the kids pools are, the adult ones are generally 'busy' but not unbearably so). We really enjoyed our port days on board.
In terms of booking on board activities we didn't book anything apart from dinner at Palo (we were on the Magic) and an afternoon in the Spa for me and DH. Both of these we booked in advance as we really wanted to make sure we could do them. I'm not sure how likely you are to be able to book those at the time/date you want if you wait until you're on board.
We found there was so much to do on board that we didn't need to book any other formal activities - we just took things as they came. Your DC are perfect age for all the character meet and greet stuff and that takes up some of each day (you get a Navigator each night with all the stuff for the next day). We also went and saw a couple of movies and saw the evening show every night. DS and DH went and did the towel folding on the spur of the moment one day. For things like mixology and choc tasting I have no feel for if they get booked up in advance or not. As it's Disney though guest services on the boat will be as helpful as they can be.
DS had an absolute blast in the Oceaneer's Club/Lab, he went after dinner every night until closing time pretty much so DH and I used to sit on the deck in the adult bar with wine. We went onto holiday time (sleep late, afternoon siesta, late dinner, late night) and it worked really well for us DH and DS had siestas, I used to go to the adult pool with a book and drink cocktails for two hours
We're doing our second cruise next year and at the moment I think I might book some more of the onboard activities for us purely because there are a few that I fancy for a first time cruise though just hanging out on the ship was enough for us and it was incredibly relaxing.
Hope this helps, if there's anything else you want to know I'll try to help.