Welcome aboard, Carly! A cruise with Disney is the best way to see the Mediterranean!
I recommend the second dinner seating, especially on the Mediterranean sailings, as many of the Port Adventures are all day adventures that might find you returning to the ship just before or even slightly after the first seating begins. Knowing you have the second seating ensures that you won't feel rushed as you return to get ready. We prefer to have more time back onboard to get ready at our own pace before heading to dinner.
If your budget allows, definitely take a Port Adventure through Disney Cruise Line in each port. Disney has set up some amazing adventures that take guests on really special trips, allowing them to experience each port like no one else can! However, I understand that the costs can add up, so perhaps you might look at the Port Adventures offered in each port and choose two or three or your must-do's and start there. Keep in mind that in a few ports aren't in the big cities, like Civitavecchia, Italy, for example, where you aren't close enough to Rome to easily get there on your own, so those types of ports are good ones to use Disney's adventures. Other ports where the ship pulls right up into the main city allow you to simply disembark and explore on your own for a few hours. However, I still think Disney's Port Adventures will allow you to make the most of your time and money no matter which port you visit.
I hope that gives you a few ideas to get started, Carly! Take a look at the
series of videos about Disney Cruise Line's European cruises that I've hosted. Keep your eye on the Disney Parks Blog each Monday for the newest video highlighting the excitement of the European cruises with Disney! Please let me know if I can help with anything else before you set sail, Carly!