don't even need필요도 없다 means “not even need” or “have no need for,” with 도 adding emphasis. The 고 ending connects this clause to another clause, often meaning “and” or “nor.”
KO: 예약할 필요도 없고 돈을 미리 낼 필요도 없어요.
EN: You don't even need to make a reservation, and you don't need to pay in advance either.