Do boxing or go boxing?

They mean different things, although you will see them used interchangeably because the outcome is the same.

• Doing something means taking part in it.
• Going something means going somewhere to do it. For example, "going swimming" means going to the pool to swim.

With your example of boxing, "going boxing" sounds more casual, and certainly doesn't sound like you're doing it at a professional level. It sounds like you are going to training sessions. Doing something doesn't necessarily mean professionally, but it sounds like you must already have the skills in order to do it.

Maybe you don't even need to use either. You could just say "I box".