Driving
Roads are okay. The city is small enough though that you could easily get around by foot or by public transportation.
On foot
There are many nice streets around, however there are many less attractive ones as well. Test your luck.
Try Nagyerdő the Big Forest: it is beautiful.
By public transport
Public transport uses local tickets. Don't try to use Budapest tickets; they won't work.
The tram (http://www.dkv.hu/) carries people to the most visited places. It starts from the railway station, goes along the main street, turns at the Grand Church a bit, goes to the Old Forest, the Zoo, Theme Park and Public Bath, then around to meet the Universities, then back. A cheap way to see most of the city is to get a ticket at the train station, get on, and enjoy the trip until you're back where you started.
Bus (http://www.hajduvolan.hu/) lines are fair. You may expect to be able to reach nearby places fast.