Welcome aboard, J! Sounds like you're planning some fun adventures as you sail with Disney!
If you book one of
Disney's Port Adventures, you won't need to worry about time zones. When you first get to your stateroom, you will find a card letting you know when and where to meet for your adventure (sometimes onboard, sometimes just off the ship in port). I'd highly recommend booking your adventures through Disney, as they have taken quite a bit of time and effort to ensure you receive the best service and experience.
Disney's ships are on the time of the port of origin. So, if you leave from Port Canaveral, you will be on Eastern Time. However, at some ports you'll match their local time. The evening before, you will find a card in your stateroom letting you know to turn your clocks forward or back. All the clocks in the ships' public areas will always display the correct time.
The best thing to do is call Disney Cruise Line and give them your specific sailing date and ship. They will be able to tell you any time zone change information you need, as it does vary by ship and sailing.
Have fun planning the rest of your adventures!