I've had a few coaches since I've been cheering, and they all did something different. My freshman year, we all waited outside in the hallway, and then went in to meet with the coach and assistant coach one by one. They told us what we did well, what we needed to work on, etc., and then if we made it, they gave us the form for camp and the practice schedule for the first month.
My sophomore and junior year, my coach had a three-day clinic-style tryout, with jumps/dances/cheer/stunting/tumbling, and then on the fourth day, everyone went and tried out in pairs for the jumps, dance, cheer, and tumbling while the coach and assistant coach watched. They would score us on all those aspects, and after that, each girl had a formal stunt tryout with a group of her choice, where she would be scored on stunt difficulty and execution. Later that night, the coach sent out an email with everyone's tryout numbers either listed under alternate or competition team. I liked this tryout method better than how I tried out my freshman year, but it wasn't exactly fair for the girls trying out for the first time winter season, because they didn't have nearly as much time to learn stunts (or set stunt groups like the girls who had done fall season).
This year, my coach had a mainly stunting-based tryout. She had everyone switching around stunt groups a lot to try new combinations, but there wasn't as much of a formal tryout. She just told everyone one by one at the end of the last day who had made the team; we didn't take any alternates. For winter season we did take a few alternates, but they've all gotten in by now because we've had quite a few injuries!
At one of my friends' schools, when girls made the team, the coach would give them their game & practice bows, which I think is a cute idea!