I charge the battery (it did this even when new.) with a charger and it starts up fine. I mow the lawn and the next time I try to start it, the battery doesn't have enough charge to turn over.

This is the way I've had to start it for a couiple of seasons now, trying different things etc.

Seems to me that the tractor is draining or just not charging the battery, but I don't know how these systems work, how a battery charges from the engine.

How can I fix this so I don't have to charge the battery every time?

It's a Scotts Lawn Tractor.

The battery is charged by a stator under the flywheel. You could have a battery that has a "dead" cell. Carry your battery and have it "load" tested.