Yes, Disney will move your luggage, but it will take a fairly long time. They don't provide rides to the new resort though...at least not as a rule. I like to just take a cab to the new resort, with my bags.
If I don't care when I get my bags, I hand them off to bell services first thing, then I go to a park. I leave that park, usually after lunch, and go to the new resort and get checked in. But, doing it that way means your bags are away from you until early evening.