Stoke-on-Trent is not really a single city — it is six historic towns strung together along a north-south axis, with Hanley at the centre acting as the de facto downtown. The towns are too spread out to walk between, so you will need either a car or the buses. The First Potteries network covers all six towns and the surrounding villages; day tickets are good value.
Local train services on the Crewe-Derby line stop at Stoke and Longton, but bus is generally quicker between the towns. Hanley's bus station is on Lichfield Street, near the John Lewis department store and the Potteries Museum. Taxis are reasonably priced and plentiful at the railway station. If you are heading to Trentham, the Wedgwood site, or the Monkey Forest, a car is much more practical.