Leuven is a small city and the historic centre is easily walkable. Most of the main sights are within 15 minutes' walk of the Grote Markt. For the outlying attractions in Heverlee — Park Abbey, the Château d'Arenberg — a bicycle is the most pleasant option.
Cycling is very much the mode of transport in Leuven. The city has an extensive network of cycle paths and the student population means that cycling culture is deeply embedded. Blue Bike, the Belgian bicycle sharing scheme, has docking stations around the city; day passes are inexpensive. Rental bikes are available from several shops near the station.
On foot is perfectly adequate for the central sights. The centre is compact and flat enough that you can cover the Grote Markt, Ladeuzeplein, Naamsestraat, and the Groot Begijnhof in a single morning's walk without any difficulty.
Buses run by De Lijn connect the centre to the outlying suburbs, but they are mainly useful for visiting Heverlee if you don't want to cycle. The main bus terminus is at the train station.
Taxis are available outside the train station and can be booked by phone or app. They are rarely necessary given the walkable centre and good cycling infrastructure.