What is the approximate knowledge of English in Slovakia? In this wiki page's chart, Slovakia places above Italy, Poland, and quite a few others. It would seem that it is not bad.
Are more young people learning English today? Can a traveller get by and make friends without knowing Slovak?
It probably goes without saying that you'll have a hundred times more fun if you spoke Slovak during your visit. Just have a go, even if it's only a couple of phrases.
To answer your question, yes the younger generation learn English and German at school as a standard. There are also speakers of Spanish, Italian and French. Also, the 30+ generation are enjoying a renewed interest in Russian (now that they are no longer forced to study it).
If it's realised that you don't speak much Slovak and the local doesn't speak much English, they may resort to German. (I'll just add that I lived in the west part of Slovakia, close to Austria.)
Outside of my English-speaking students, I found that the chances of you finding an English speaker are fairly good, but in general most who claim to speak English are either too shy, or lack enough knowledge for practical use. Of course, it's easier to use English in a larger city/town. OK, I think this is pretty much the case with a common foreign language in any country, but it's a good reason to bone up on Slovak anyway.