Kőszeg's old town is tiny and entirely walkable. The train station is a ten-minute walk from Jurisics tér, the main square. You will not need any transport to see the historic core.
On foot: The castle, the main square, the churches, and the town walls are all within a few minutes' walk of each other. Comfortable shoes are recommended; the streets are cobbled.
Cycling: The town sits on well-signed cycling routes that run along the Austrian border region. Bikes can be rented at a few places in town. The terrain around Kőszeg is gently rolling for the most part, with more serious climbing if you head up into the Kőszeg Mountains toward the Írott-kő summit.
By car: A car is useful if you want to explore the surrounding villages, drive up toward the Írott-kő trailheads, or visit the wine cellars in the hills above town. Parking in the old town centre can be limited; there are free spaces just outside the old walls.