The most important international cycling track in Slovakia is Eurovelo 6, which is connecting the Atlantic Ocean with the Black Sea. Most of the track follows the Danube River. If you take this track, it will safely lead you to Bratislava through the fields of Marchfeld and bottomland Danube forests from Vienna (70 km, flatland), Komárno (110 km) or Budapest (260 km).
Track signing of another European cycling track Eurovelo 11 is currently being undertaken in the East Slovakia. It leads from the north of Norway to Athens. Slovakian part of the track goes from Pieniny mountains through Prešov and Košice to Hungary. But you can find many not really frequented, quality tarmac roads off from the European cycling tracks.
More on cycling in Slovakia