For centuries Slovakia was part of the Kingdom of Hungary, and therefore a province of the Habsburg Empire. In 1918, after the first world war, it was combined with Bohemia and Moravia to form the new Republic of Czechoslovakia, which subsequently, in 1993, was split into two separate states: Czechia [the Czech Republic] and Slovakia. The capital of Slovakia is Bratislava (in German Pressburg).
The most famous bike route, and the only one documented by yours truly, is the Donauradweg which crosses through Slovakia for a short segment. As of 1995 this segment was paved in good condition.
Bratislava, at the time of my one bike visit, was a city hardly considered bikable, the only true bike path being the aforementioned Donauradweg. Since then the situation should have improved somewhat.
Slovakia is now part of the European Union and has adopted the Euro as its currency. Therefore there are no problems with currency exchange, at least for other citizens of the European Union.