The town square is the right place to start. The half-timbered houses ringing it date from the 16th and 17th centuries, and the Town Hall on the north side — built in 1552 — is one of the oldest functioning town halls in Denmark. Sit outside one of the cafes and just look around; the architectural coherence is remarkable.
On Wednesday and Saturday mornings a proper outdoor market sets up here. It is exactly the kind of market where you buy things you did not know you needed — smoked eel, hand-thrown pottery, pickled herring in a dozen varieties. The square also hosts the annual Køge Jazz Festival in August, which draws big crowds from Copenhagen.