Everybody has their own way, but mine is to water well (until water comes out the bottom of the pot) then wait and allow sun and wind to dry the soil out until it's dry or almost dry at depth. I know from experience how fast the different size pots dry out here, given the exposure, so I start cuttings in a pot that I know will dry out within my regular watering window (which is about once a week). The critical thing is the depth of the pot (more than the width). If you use a shallow pot up front (like azalea pot proportions) that gives you greater security about the bottom not staying wet. Or you can just fill a deeper pot up partway and get the same effect.
I would recommend leaving your cuttings where they are, if the pot is not real deep. There's no sense in moving them up a size until you know they have roots (either by seeing the roots or usually by seeing new growth on top). Tiffany makes a good point about these not being your usual desert cacti. Both kinds can be accommodated within the boundaries I have described (a watering cycle consisting of thoroughly wet to mostly dry, each time).
