For the moldings, you'll need to remove the screw/rivet/whatever that holds the rear edge of the molding to the inside of the door, and then carefully pry the molding off of the door.
I removed the remaining adhesive by using a heat gun to loosen/release it, and then just slowly working it loose with my thumb (a plastic putty knife can be helpful here, too.) It took a long time, but was worth the effort. The last few bits that stay behind can be removed with some rubbing alcohol (I'm sure that stronger adhesive remover would also work, but I didn't find it necessary to use that on this particular stuff.)