If I am playing in a tournament and I win the first game of the round, when does my opponent decide if they will play first? Does it happen before side boarding or does it take place some time after presenting your deck for shuffle? If REL matters please specify the differences at various REL.
You must decide before looking at your first hand of the game, it can be after sideboarding. If you do not choose it is assumed you will play.
From the Magic Tournament Rules 2.2:
For the first game of a match, the winner of a random method (such as a die roll or coin toss) chooses either to play first or to play second. The winner must state this choice before looking at his or her hand. If the winner states no choice, it is assumed that he or she is playing first. The player who plays first skips the draw step of his or her first turn. This is referred to as the play/draw rule.
After each game in a match, the loser of that game decides whether to play first in the next game. They may wait until after sideboarding to make the decision. If the previous game was a draw, the player who decided to play or draw at the beginning of the drawn game chooses.