After leaving school, Terry studied English at St Edmund Hall, Oxford University. Although he was strongly discouraged from acting by his school headmaster, he became involved in drama at the University. In his final year he co-wrote a revue about capital punishment called Hang Down Your Head and Die , on which one of the other writers was future writing partner, Michael Palin.

Moving from Oxford to London at the end of his degree, Terry started working for BBC Light Entertainment, where he gradually started writing sketches. Eventually, he and Michael ended up working on The Frost Report , an influential comedy programme of the 1960s, for which John Cleese, Graham Chapman and Eric Idle also wrote material.