How many times a watch can be professionally polished without getting into altering the thickness and/or shape really depends on how aggressive they've been. A watch going into service without dings won't require much beyond a very light superficial polishing. However, one with more surface damage will need more aggressive polishing in order to successfully remove them. That's why many people just live with certain "character marks" and chalk it up to telling a story instead of having them removed all the time. Going after absolutely everything will likely take its toll as the years roll on.
A Cape Cod polishing cloth is super easy to use. Just remove the strap and rub the cloth in a back and forth motion, applying a reasonable amount of pressure. Then buff off the residue with a soft microfiber cloth. Just stay away from all brushed surfaces, as it will both change the grain and make it shinier than it was originally. Other than that, you're not removing much with the cloth and can erase away little swirl marks without much effort or fear.
Have fun with it!