The answer requires knowing only 1 piece of information: who served first in the set that went to the tiebreak? The other player will always serve first in the next set.
Why? Connect 2 simple facts.
Fact 1) The player that served first in this set will necessarily be the first to serve in the tiebreak.
Fact 2) The player that serves first in the tiebreak, irrespective of the tiebreaker outcome, will return (i.e. NOT serve) in the first game of the next set.†
Combining fact 1 and 2: If you serve first in a set that goes to a tiebreak, your opponent will ALWAYS serve first in the next set.
KISS: Keep It Simple Stupid
A lot of other explanations make it so confusing. It's easy: The player that serves first in a set that goes to tiebreak, will never serve first in the next set. The outcome of the tiebreak has nothing to do with this. You only need to know who served first in the set that went to a tiebreak to know who will serve first in the next set. It is always the OTHER player, always.
† Some people ask why fact 2 is true, which is an entirely different question. But it is worth addressing.
A simple rule to remember is that Players can never serve in consecutive games. Starting at 0-0, once you get to 6-6, it is necessarily the turn of the player who served first in the set to then serve first in the tiebreak. And since the tiebreak is considered a service game in-and-of-itself, and is served first by the same player who served first in the set, the next game after the tiebreak is the first game of the next set. And this must be served by the other player.