The main reason to avoid paper towels, tissues, and etc. is that they shed lint (paper fibers) which will be left behind on your slide and eventually interfere with your prep. They look small to the naked eye, but they are not small compared to the prep on your slide.
The other issue with paper towels and etc. is that there might be some particulate matter in the paper that's hard enough to scratch glass or optical coatings. This is more of an issue for optics - especially coated ones - than for cheap microscope slides, but it's something to consider.
If you're not mounting permanent preps then ultrasonic cleaning is probably a bit of overkill - although I have a surplus sonicator and it does a great job on a lot of tasks. It's generally sufficient to rinse the slide with distilled or deionized water and then follow up with (clean) alcohol. Optical purists will say spectroscopy grade methanol, but for home use any reasonably pure MeOH, EtOH or iPrOH will probably do. The main concern, again, is leaving behind particulates or residues. You can allow the slides to air dry if you're using pure alcohol, or use a piece of lens tissue (lint free paper tissue) to help them dry faster. I've also used Kimwipes in a pinch, but they're not truly lint free so not ideal. They're still much better than a tissue or paper towel.