Who serves after a tiebreak?
Whoever received the first point of the tiebreak serves the first game of the next set. Change ends before it: 7-6 is thirteen games, and ends change after every odd game.
The check that settles it at the net
Think back to the first point of the tiebreak. Whoever served it does not serve now. That is the whole rule.
It reads oddly until you remember that a tiebreak is a game like any other, and at the end of a game the server becomes the receiver. The tiebreak just happens to be the thirteenth game of the set.
You change ends as well
A set won on a tiebreak finishes 7-6, which is thirteen games. Ends change after every odd game, so you swap before the first game of the new set.
Sets that end on an even total do not work that way. After a 6-4 or a 6-2 you stay where you are and change ends at the end of the first game of the next set instead.
In doubles
Same answer, with one extra choice on top. The pair who received the first point of the tiebreak serves the first game of the new set, and they pick which of the two takes it. They are not tied to the order they used in the last set: Rule 14 hands every pair that choice at the start of every set.
And after a match tiebreak?
Nobody serves. A match tiebreak ends the match, so there is no next set to open.
In the rule book
ITF Rules of Tennis, Rule 5b, Tie-break game:
The player/team whose turn it was to serve first in the tie-break game shall be the receiver in the first game of the following set.
Common follow-ups
Who serves the first point of the tiebreak itself?
Whoever was due to serve the next game. At six games all the serve already belongs to somebody, and that player takes point one. The other side then serves two, and it alternates in twos to the end of the game.
Does the order inside a doubles team change during the tiebreak?
No. Inside the tiebreak the rotation within each team stays in the order it was in during that set. The choice comes at the start of the next set, not during the game.
We got it wrong and played a whole game. What now?
Rule 27 says the points already played stand. In an ordinary game, under Rule 27c, spotting it mid-game means the player who was due to serve takes over at once, and a game that finished before anybody noticed leaves the order as altered. Inside a tiebreak Rule 27d applies instead, and the test is a different one: an even number of points played and it is corrected immediately, an odd number and the order stays as altered.
Luasc hands the serve over at the end of the tiebreak and shows who has it and which half they serve from, so nobody has to reconstruct point one. What else it keeps track of.