The only buffing pads I know of where the manufacturer actually recommends throwing them in the washer and dryer are Meguiar's Soft Buff 2.0 pads - the ones with the velcro backing laminated to the foam, not just glued on. Lake Country and most others caution against machine washing as doing so can fairly quickly break down the adhesive and cause the velcro backer to come off.

For those pads either use the Dawn dish detergent in a bucket of water, or a little degreaser in a bucket of water. I know a lot of guys who use Meguiar's Super Degreaser from our Detailer Line for this - diluted 10:1 with water, and then thoroughly rinse with water. But since most compounds and paint cleaners tend to be water soluble anyway, the degreaser is probably over kill. Dawn will work very well, just make sure to get it all out.

You don't want to wring out the pads as the twisting motion isn't a good thing for the foam structure, but squeezing them is fine. In fact, if you lay a pad onto a clean terry cloth towel and then very tightly roll it up in said towel, you can remove the vast majority of the moisture. After that, stand them up on end leaning in toward each other in groups of three (like making a little tee-pee) and allow them to air dry. I like to do this with the face of the pads turned toward the inside of the tee-pee so dust doesn't land on the face of the pads while drying. Or I'll lean 'em up against a bottle of product to do the same thing. Since it's so dang dry here in SoCal they usually dry overnight, at which point I stack them in a cabinet with plenty of airflow around them to both keep out the dust and prevent mold.

For the record, this is the exact same way we handle our pads here at work. And we go through more pads in a week than this whole forum probably goes through in a year!!