Hi Shannon,
Disney's Caribbean Beach Resort is served by bus transportation. There is bus transportation to all of the theme parks, water parks, and Downtown Disney area from your resort. On the day you have a reservation for
Chef Mickey's, I would take a bus to the Magic Kingdom Park and then walk or take the monorail to
Disney's Contemporary Resort. Even though walking sounds like it would take longer, it can be much quicker. If you happen to have a very early reservation for breakfast, you might want to check with the lobby concierge at Disney's Caribbean Beach Resort to see if there will be an early bus for the Magic Kingdom Park. If we are particularly tired from late nights at the parks, we have been known to take a taxi to an early reservation simply because it saves time and we get to sleep just a few precious moments longer!
After breakfast you can walk to the Magic Kingdom Park or take a ride on the resort loop of the monorail. At the end of your park visit, you would return to Disney's Caribbean Beach Resort by bus.
Wishing you a magical vacation,
Amy H