Tarsus Museum is the best indoor stop in town and the place that ties the scattered ruins together. The collection covers the city's long run through antiquity, early Christianity, Islamic rule, and the Ottoman period, with inscriptions, sculpture, coins, household objects, and material from nearby excavations.
Allow more time than you might expect for a small-city museum. Tarsus has a deep archaeological record, and the museum gives useful context before you go looking for Roman road paving, old houses, and the St Paul sites outside.