The park itself needs no reservation, but you need a tour ticket to go into the cave.
Backcountry camping permit: Permits are $10, online on Recreation.gov or in person at the Mammoth Cave Campground kiosk.
Cave tour tickets: Tickets are required for every cave tour. Reservations are strongly recommended and are the only way to be sure of a spot. Book on Recreation.gov or call (877) 444-6777.