Just inland from the center of Demre are the remains of ancient Myra. Some well-preserved Lycian rock tombs are hewn into the cliff face, and there's also a large Roman theater, which has been wonderfully restored.
It's an atmospheric site, with sculpted masks chiseled into rock leading up to the theater and the rock tombs looming above.
Combining St. Nicholas Church and Myra together in a trip is an excellent day out.