At the resort you're leaving, be sure to call bell services early to make an appointment to have your bags picked up. Remember, everyone has the same 11:00 check-out time, and things can be a little hectic in the mornings. We always call the night before to set up a time with them.
At the new resort, it doesn't work like Disney's Magical Express -- the bags won't be put directly in your new room. You have to either go to bell services or call them and ask to have them delivered.
It costs you a tip at both resorts, but it works very smoothly.