Yes, definitely repot them soon. Ceramic pot is better than plastic one and mineral substrate is better than peaty. You will just have to add diluted fertilizer sometimes.

Nice plants. Can't help with identification but I've repotted many cactus recently to a new growing medium.

You will want to pot these in a fast draining mix. Many members here like to make there own gritty mix (you can search the forum, many threads) or you can amend a commercial bag of potting soil by amending it at least 50/50 or more with something such as pumice or perlite. Sand may also work as long as it's very course, not the type of sand used in sand boxes. If possible, use a potting soil without peat/low in peat as once it dries out it's nearly impossible to re-wet and ends up just repelling water. I would recommend avoiding top soil.

You can pot these in ceramic just fine as long as there is a drainage hole in the bottom. Before potting, clean as much of the old substrate as you can from the roots and let the plant rest in a shady, dry spot for a few days to a week; this gives the roots time to heal after cleaning. Once potted wait a few more days before you water.

You'll want to water your plants once the soil dries out during the growing season, typically spring through fall with most cactus. The timeline can vary depending on the plant's environment, pot and medium but about once a week seems typical for me. I check my plant's soil before watering by sticking a wooden skewer into the soil along the edge of the pot all the way down, if it comes out dry the plant is ready for water. Water thoroughly until water comes out the drainage hole and make sure all excess water is allowed to drain out the bottom of the pot.

During dormancy, typically winter and sometimes the height of summer as well if it's really hot, these plants will need very little water if any at all as they won't be actively growing. A good rule is if in doubt about watering, don't.

Good luck with your new plants!