Make sure you have potting soil that is specifically for cacti. Desert plants need well-draining soil. When re-potting plants chose a container that is one size bigger than the current container.
It is better to under pot than over-pot. Cacti are slow growing, low maintenance plants and only need to be re-potted every two-three years. Choose an un-glazed pot so that the moisture can be wicked away from the plant.
Terracotta is a good choice. The best indicator that it needs to be re-potted is when the roots become root-bound. If you see roots coming out of the drainage holes it is a good indication that the roots are bound.
You should wait about a week after repotting cacti to water them. The number one killer of cacti is overwatering. Let the soil completely dry out between watering.
Remember, they are desert plants and can go many months in hot dry weather without being watered. Or, you can take my dad’s advice and only water your cactus when you see on the weather channel that it is raining in Mexico (seems legit!)