The Rynek has a handful of cafes with outdoor seating in summer, and the atmosphere is low-key and local. You'll find the standard Polish cafe setup: good coffee, cake, maybe a beer in the evening — nothing that demands its own trip, but perfectly agreeable after a morning in the cathedral and museum.
Don't come to Gniezno for nightlife. This is a pilgrimage city and a day-trip destination, not a party town. The evening quiets down early and that's fine — there are worse things than a peaceful square and a beer after a day of history.