Say I somehow make an illegal move, what happens? If I realize, can I then take it back, and make it legal? If so, how?
It is possible for a position to change in such a way that an illegal move becomes legal. As one example, if a move would result in your king being in check, then it is illegal, but if you make one (or more) move(s) so that later the same move would not result in check, then it becomes legal. The following diagram presents one such situation:
[FEN "q1k5/8/8/8/8/8/1Q6/1K6 w - - 0 1"] [StartFlipped "0"] 1.Qa2 Kb7 2.Ka1
In the beginning position of the above construction, the white king cannot move to a1, since that would be check. However, if white first moved their queen onto a2, then they would be able to move their king to a1 later.