We booked our excursions through the cruise line. Generally we have done this, however many people have great success in booking a private excursion or picking up a taxi in port.
What you wish to do will depend on your family group and what you may wish to do. If it's your first cruise and your first trip to the Caribbean, honestly my recommendation would be to stick with the Disney organized activities. The one comfort is that the company providing the excursion will be guaranteed to get you back to the ship in time -- often in plenty of time, LOL.
You will be able to book excursions, spa appointments, adult dining, character breakfast and more. Online check-in is to establish a port arrival time, which dictates what time you are able to board the ship.