What's the best way to sand down wooden shutters for painting?

Can you take them apart and sand the pieces individually? That would be my choice. You can also correct any warping or wear while they're apart, as well.

Make sure you are using the right grit. I would take the shutter to your local home improvement store and they can help you determine which is your best bet.

I've done louvered doors in the past, and I sanded and then used a foam brush to paint. I suspect a spray painter would be much easier.

Since you have painted the shutters before, and you want a smooth result, you will have to go all the way down to the wood. You probably need to start with a good cleaning, too. Sand, pollen, bug bits and other things can make your shutters dirty and bumpy. When the shutters are clean and sanded (and cleaned off again), I suggest staining them. There is even white stain, as well as wood stains and other colors. Some paints are stain and seal, so 2-3 coats, front and back, should be smooth and protected for a couple of years. Best wishes 😇

palm sander on most then sandpaper