Replacing wet or dry cloths is done the same way - you simply pull off the used one, toss it, and replace it with a new one. There are four "push holes" (for lack of whatever the correct term is) and the cloth simply wraps around the mop face, and your press the four corners of the cloths into the holes. Done and done!
The starter set comes with wet and dry refills, though in my experience I tend to use the wet cloths far more than the dry. Dry works well, though, for my hardwood floors. Wet, I find, works best in my kitchen on my Travertine - especially after a spill, and even moreso if it's something sticky, like soda.
For whatever small bit that it's worth, I do make every effort to clean spots - esp in the kitchen - with a wet rag and a good "green" cleaner. I definitely wouldn't recommend using the Swiffer as your new scrubber - it's not designed for that kind of cleaning. The Swiffer, I find, is more for the general cleaning days, and just keeping the house or apartment floors clean. Lesson learned: you can also end up wasting a lot of Swiffer cloths - and money - by trying to use the Swiffer like a scrubber. Yep, some things you still have to do the old-fashioned way, with a little elbow grease :)
I shot a quick little video - Im not terribly familiar with making videos, so I hope you'll bear with me, and I hope it helps... :)