Twisty Puzzles a la Vi
is an excellent video to help provide an overview of what to expect when making puzzles. Eitan talks about using the software and making designs, but briefly. Mainly, he walks you through himself making a puzzle from Solidworks design to printed, dyed, and stickered puzzle.
Mainly, the search function on this forum as well as in your preferred search engine can be really helpful. Solidworks is one of the names I hear a lot, but others like Rhino are used too. Solidworks has a ~$150 student edition which does make it far more available. Before buying any CAD software, I suggest using a trial version to see if designing is really for you.
Andreas Nortmann wrote:
Things like this are illegal.
If not I will pass an appropriate law.