Screws first, in the close handle and the release handle pockets, then work around the perimeter carefully. I know on my Mom'* LeSabre, there was a hidden screw mine didn't have, and I found it by accident, so pry carefully, and 'feel' for anything you may have missed.
There'* a special tool for pulling out the little plastic screw/thingies. You can pick one up at Auto-Zone cheap, and it'll keep you from accidentally cracking the panel by pulling too hard.
You slide it between the metal of the door and the panel until you find the fastener, slide it around the fastener, then pry it out. It pulls against the fastener itself instead of stressing the panel.