If your shutters are sagging, take a close look at the point where the frames are attached to the walls. Are the hinges rusted, loose, or unable to hold onto rotting wood? As you carefully remove the shutters from the wall with a screwdriver (save any hardware you remove), look for clues as to what's letting gravity have its way.
If the window frame or wall suffers serious rot or water damage, you've got a bigger project on your hands. You'll need to identify and eliminate the root cause, replace any rotted wood, and protect new pieces from the elements. If the hinges have simply worked themselves loose, you're in luck; we'll tell you later how to tighten those hinges to stay.
To begin, protect your back by making sure your work area is at least the height of your hips, and put your tools within easy reach. Remove the hinges from the shutters and work carefully through each piece. Disassemble the hardware only as much as needed to get a good angle on sanding and begin prepping your surfaces for paint.
A power drill with a wire brush or sanding attachments can definitely save you time. When sanding, work with the grain of the wood and use a light touch to avoid gouging the underlying surface. The best results will likely come by using a succession of fine sandpapers. Detail sanders are also great for this kind of work, and come in a variety of attachments. They range from $120-$200.
Planning on staining? To bring out the grain, you'll want to strip the shutters down to bare wood. Depending on your shutters, this can be quite involved; plantation style is the most difficult. Chemical stripping is definitely the most expedient and can also be used to remove paint.
Chemical strippers are available at your local home store (be sure to follow all directions and cautions about vapors, skin contact, etc.). You might also consider having them dipped and stripped by a pro. This will definitely save you time and exposure to nasty chemicals, though these services do not come cheap.
Jane Tip: If you're repainting more than 3 to 4 shutters (especially if they're plantation style) you might look in to renting a spray gun for the day; it'll cut you painting time by about 90% and will leave behind a light, even coat of paint.
Jane Tip: Make a shim with a small piece of wood and gently tap it into the existing screw holes. This will give the screw something to grasp onto and will tighten the hinge.
Your Shutters, Saved!
Stand back and admire. Your shutters are now good for another ten years as solid sentinels against Mother Nature.