The rulebook says the player that has the advantage is the first player and acts forst that round. But if the first player is the player without the momentum token, does the first player get the momentum token? I can't find anyplace in the base rules that says when/ if the momentum token changes hands.

The scenario details tell you which player has the momentum on the 1st turn and thus takes the first action. If you don't use the Gambit Cards, then during the Rally Phase of all subsequent turns, you determine advantage by seeing which player has the most unused Command Tokens left. In the case of a tie, the player with momentum retains it. Playing a Gambit Card when you have momentum, gives your opponent momentum next turn. Cheers!

In the Core Set, momentum never changes hands. If using Gambit Cards, the player with the faceup Gambit Card has Momentum. The scenario only indicates who starts with Momentum, i.e. the player who will start with a faceup Gambit Card.