For some reason I'm under the impression that the players have to pay for their own equipment (bats, gloves, cleats, etc...). But all that stuff is probably provided under endorsement deals anyway.
Players at the Major League level don't have to pay for their equipment. Most minor leaguers don't either. It's only in the very low minors that a guy might have to buy his own glove or spikes. It's not provided by the team, but as you say, it's generally provided by manufacturers. That said, they do get a 1099 for such equipment and have to pay taxes on it as if it were income.
Most players also pay things like gym memberships, personal trainers, etc. That comes out of pocket, but it's also tax deductible as a business expense.