Do not mess with the roots unless you see a good reason to do so. That means leave them in the root ball they are currently in, without breaking it up or injuring them in the process. You create a risk of rot when you do this, and it's best avoided unless there is some gain.
Skip the rocks at the bottom of the new pot. They are not serving any useful purpose. The important thing is to use soil with good drainage. Like a premade cactus mix, or regular potting soil plus perlite/pumice/whatever to about 25-50%.
It's very easy to repot your plant without causing problems. Just choose a pot that's not a lot bigger than the one it's in, and wider than deep. Remove the plant carefully from the pot it's in (without handling the roots more than necessary) and place it on top of soil in the new pot, then fill around the edges. Wait a week to water afterwards. There is very little risk in this approach if you're careful and the pot is the right size.
Be aware that unglazed clay pots behave different from regular plastic ones with respect to watering. Because the water can exit through the container from the sides, as well as from the top like normal, the soil in an unglazed clay pot will dry out faster than normal, especially in the sun, and especially in warm, dry conditions. So you might need to water more often than you're used to, but try to exercise restraint and wait until the soil is drying out at depth before watering.
