Czechia (the official name Czechia has not caught on among English speakers, who seem to prefer the more cumbersome name Czech Republic), created in 1993 from the partitioning of Czechoslovakia, which had is own creation after the “Great War” [World War I] by combining two areas of the dismembered Austro-Hungarian Empire, is trying to establish a network of bike routes similar to those of neighboring Germany and Austria.
There already exist decent mapping and decent signage (signs are yellow or white); the problem is that all too often bike paths that are indicated as cyclable, in reality are merely dirt footpaths or paths that have surfaces so broken-up that they would be more appropriate to the “Hell of the North” that professional cyclists face every year in the Paris-Roubaix road race.
Therefore, if you wish to visit this country on bike, unless a you are a mountain bike enthusiast, you would do better to use local and secondary roads, which generally have little traffic, rather than the routes designated as bike paths. The bike maps that I relied on make no distinction between paved paths, or smooth in any way, and those paths paved in cobbles or even in sand.
One scenic route of particular interest that I tried is that from Prague following along the Moldau (on a very bumpy bike path) as far as the confluence with the Elbe near the beautiful city of Melnik; from here following the Elbe on low-traffic streets, and after Decin, on a real bike path, one reaches Germany and can go as far as Dresden.
Bike routes are indicated with signs that mostly are yellow or white. There are decent bike maps available; in particular ShoCart is a map at 1:75,000 which covers the entire Czech territory.