The centre is compact enough to walk. Beyond that, the bus network run by local operators covers the city and surrounding villages well. An OV-chipkaart (the national public transport smartcard) works on all buses.
Cycling is the most popular way to get around, as it is everywhere in the Netherlands. The city has excellent bike infrastructure and rental shops are easy to find near the station. For exploring the Veluwe, a bike is by far the best option -- the park has a dense network of numbered junction routes you can follow with a simple map.