What is the correct way to say that a team got to the next round? For example,
"Team A won the quarter finals and got (the correct verb here) to semi-final"
It qualified for the semi-finals.
1.1 Become eligible for a competition or its final rounds, by reaching a certain standard or defeating a competitor: (Oxford)
England are in danger of failing to qualify
Or the team advanced to the next round.
In UK English, you'd often find the word "Progressed" used to describe moving onto the next stage of a contest
Southampton progressed to the FA Cup fourth round after a narrow 1-0 win at Ipswich in their third-round replay at Portman Road.
I'm personally more inclined to use Tim Romano's suggestion of advance, but another term worth considering is survive, e.g. "The team survived the first round," or "The team survived to the final." This term emphasizes the struggle of the competition and might be considered more dramatic.