Central Limerick is walkable: from the medieval quarter on King's Island to the southern end of O'Connell Street is about a kilometre and a half — 20 minutes at a steady pace. Most visitors do everything in the centre on foot.
Local buses, operated by Bus Éireann and TFI Local Link, cover the suburbs and the route out to the University of Limerick. The standard fare is paid in cash or with a TFI Leap Card, which can be picked up at the Colbert Station newsagent. Free TFI bikes (registered through the app) are scattered around the centre and are useful for the longer walks to Thomond Park or the university.
Taxis are abundant and inexpensive by Irish standards. The main rank is on Sarsfield Street, but Free Now and the local Bunnycabs app both work well. Driving in the centre is unnecessary and parking is metered; the multi-storey at Arthur's Quay is the most central if you do arrive by car.