We have driven the last two years in a row from Grapevine...
Both times we have taken the exact same route that JVCJen descibed.
Two years ago we drove straight through... seems like it was right at 20 hours.
We got in late in the afternoon. Really a wasted afternoon and motel night (stayed at All Star Movies). Really, really tired and grumpy when we arrived... didn't do anything the first night.
Last year, we drove to Tallahassee and stayed. We left very early in the morning and got into Talahassee late in the evening (if I remember correctly, it took about 14-15 hours). Then we got up after sleeping in, and drove into Orlando (only about 4 to 5 hours). When we got to the room (Pop Century) we weren't nearly as tired. We still didn't go to a park the first day, but enjoyed our time relaxing by the pool, planning our next morning, etc. We started out the next day first thing with a character breakfast at Chef Mickey's and it was great.
We are going again this year in late March... spring break. When are you going? We are going to arrive Friday, March 18 and coming back March 24 (leaving here on Thursday the 17). We are going to try the Port Orleans French Quarter this time.
Hope all of this helps.
Have a great time!