A coach is somebody who helps a person or a team to improve their technique or style in a sport. A trainer is somebody who teaches the skills needed for something. An instructor is somebody who helps a person learn a new sport.
The coach was happy with the performance of the team.
Our coach is always complaining about our lack of style.
She works as a teacher trainer (=She trains future teachers).
He’s a dog trainer (=He teaches some skills to the dogs).
My skiing instructor was happy with my performance.
The driving instructor couldn’t believe that she didn’t know how to start the car.