In the simplest case you can repot by removing the plant with the entire root ball intact, soil and all, and dropping it into a bigger pot with more soil. If there's no root damage (no disruption of the root ball) in the process then you can water right away or whenever it's convenient. I have heard advice to separate and prune roots but never seen the need to do it. The more you manhandle the roots, the longer you should wait until watering. Up to about a week or two.
If you repot regularly (to the next size up but no bigger) then there's no need to prune or mess with the roots. Better left alone really, in the short term anyway.
Leaving succulents out of soil is something you might consider after taking a cutting, or if there's some kind of bugs you want to get rid of down there. As a general rule you can bare-root agaves and leave them out of the soil (in the shade) for a week or two. Most other kinds of succulents would prefer to be in the soil after you separate them or whatever. Regular potting soil is a bit moist (not bone dry anyway) and that's enough to keep the roots from desiccating until you water.
