Esztergom's main sights are all walkable from each other. The train station is about 15 minutes on foot from the castle hill. From the station, walk south through the town centre, then climb the hill via the Dark Gate in the Víziváros district.
There is a small local bus network, but it is not particularly useful for tourists. Taxis are available near the station and the main square. Bicycles can be hired in summer — cycling along the Danube bank and across the Maria Valéria Bridge into Slovakia makes a good half-day excursion.