I finally got around to repotting a couple of cacti that I own and decided to take some photos to detail how I repot a cactus. Normally I would recommend repotting a succulent during the early spring when it enters the active growing cycle but I didn't get around to it and there are still plenty of warm days ahead.
The first thing you should do when repotting a plant is to gather your plant, pot and soil in the area you will be working in.
Choose the right size pot for your plant. In this case the cacti is small and isn't a very fast grower so I chose a pot just slightly larger. If you choose a pot that is too large your soil will remain wet longer which could be detrimental to plants like cacti.
Lift the cacti from it's pot. You can use a thick pair of gardening gloves or tongs if you are worried about getting pricked by the spines. I didn't have any handy so I made a strap out of newspaper to lift the cacti and not get poked by it. Take this time to examine the roots and make sure you don't have any mealy bugs living in the soil. If you look closely at the top right photo you'll see that this cacti was severely root-bound, so I broke up the roots a little to encourage them to spread in the new soil and pot.
Add a small amount of soil to the bottom of the new pot and place your cacti inside. Check to see if the bottom of the plant is level with the rim of the pot. If it isn't lift the plant and add more soil until it is level. Finally, fill in soil around the plant making sure that soil fills in the pockets created by the roots. I like to stick a bamboo skewer in the soil around the plant and jiggle it to make sure the soil settles in. You can also tap the sides of the pot or lift and swiftly (but gently) tap it on a firm surface to make the soil settle in between the roots.
Its not rocket science but maybe this will be helpful to someone out there.
Related: How to Repot Lithops "Living Stones."