What does Que les gusta hacer a Pedro y a mi mean in English?

First, the question should be "que nos gusta hacer a Pedro y a mi" because the "a mi" is implying yourself therefore a nosotros construction must be used.

Gustar is a verb that means "to like"

Nos gusta would mean we like. Hacer is the verb that means "to do".

So to answer your question it means "What do Pedro and I like to do?"