The historic centre of Győr is compact and almost entirely pedestrianised — you can walk from the train station to the cathedral in about 15 minutes, and the main sights are all within easy walking distance of each other. For most visitors, feet are the only transport needed.
The city has a bus network for the outer areas, but unless you are going to the Audi factory on the outskirts or exploring the residential neighbourhoods, you are unlikely to need it. Taxis are available and reasonably priced by Western European standards. Cycling is popular: the riverside path along the Mosoni-Duna offers a flat, pleasant route and bike rental is available near the main square.