I love living so close to the Czech/Slovak border. A few weeks ago, I went hiking along the ridge that separates the countries, and I went to Slovakia about 50 times, hopping back and forth like a kid over the border markers.
This is from the red trail west of Kohútka. That thing I'm leaning on is one of the many old border markers. The "S" stands for Slovakia. There's a "C" on the other side for the Czech Republi, and a line through the top marking the exact border. Some of the lines even change angles right on the stone. Very cool.
If you go...
If you want to go without a car, I recommend getting a bus to "Nový Hrozenkov,Vranča,konečná" and then hiking up to Kohútka via the yellow/blue trail. Stop at the restaurant there for lunch or a snack. Consider their famous blueberry knedliky (a sort of dumpling filled with blueberries and drenched in cream, butter, and sugar). Then hike the red trail west as far as you like before heading down to a road or train line. I took the blue trail to Huslenky (well, I got lost on the blue trail, but ended up in Huslenky anyway). Great hike.