The main software you will need to make maps is
Zone Builder, using the game configuration found in the "Download Now" section of the
SRB2Kart release post (almost at the bottom).
While not necessary, you might also need
SLADE if you plan on adding textures, packing several maps in one file, or just about any wad/pk3 related operations done the right way.
Some map making resources like tutorials(?), tips and tricks, how to set up specific things like checkpoints and such, and what-does-whats can be found in the SRB2Kart Discord server (the invite is at the top of the SRB2Kart release post).
The
Zone Builder tutorial in the wiki will be useful for setting up Zone Builder and learning how to use it a bit, as well as some terminology.
While the tutorial is oriented for SRB2 itself, SRB2Kart mapping is largely similar to the original game - Kart just adds some effects and objects you can use. Make sure you set up the SRB2Kart configuration file right.
Have fun!