Hi, Jean!
Isn't planning a Disney Cruise Line (DCL) vacation so much fun?! There are so many things to do and see...on and off the ship!
Since I'm not completely sure of your itinerary, I'm going to answer your questions in a couple of different ways, and if I totally "miss the boat" (pardon the pun), let me know!
For
Castaway Cay (Disney's Private Island), there is a tram that transports guests from the ship to the main guest area of the island. However, it's not an incredibly long walk, and my family and I have always enjoyed walking. Once you're in the main area, you can rent bicycles to get around the island, and there are several shore excursions/port adventures from which to choose. You can certainly do several in a one-day visit to Castaway Cay. A one-day stop on Castaway Cay is usually from early morning until around sunset -- a really good day in paradise!
As far as visiting other islands' ports of call, there are numerous options in terms of transportation and shore excursions. The Planning Center is a great place to start when trying to decide what you would like to do at each port and how much fun you can squeeze into one day. For me, I have a healthy respect for the ship's schedule. Therefore, I like to book my Port Adventures through DCL knowing that if something happens to our transportation (a bus breaks down/traffic issues), the ship will be notified and will wait. Otherwise, if I book separately and take a taxi or bus and something happens, I am on my own! Plus, you certainly don't want to overbook your day and have difficulty trying to get from one adventure to another and back to the ship on time.
I hope that helps you plan. Let me know if I can answer more specific questions. Have a wonderful day and a magical vacation. Bon voyage!