Enter the lists is an idiomatic expression meaning to participate in a struggle or competition. As far as synonyms go, you could say that someone is "throwing their hat in the ring," "joining the fray," "entering the arena," or "stepping up to the plate." Other phrases like "taking the plunge," "diving in," "or "getting involved" could also apply, depending on the context. Essentially, any phrase that conveys the idea of someone willingly taking on a challenge or assignment could work as a synonym for "entering the lists".