Which two schools of philosophers in Athens began debating with Paul?

Answer: Epicureans and Stoics

Some Epicurean and Stoic philosophers debated Paul; some called him a babbler because he preached Jesus and the resurrection.

“Some Epicurean and Stoic philosophers also began to debate with him. Some of them asked, “What is this babbler trying to say?” Others said, “He seems to be advocating foreign gods.” They said this because Paul was proclaiming the good news of Jesus and the resurrection.”— Acts 17:18 (BSB)