Hello goldenecho!
I've been asking a few NA friends who know Slovakia to see what they may find important to know for their countrymen.
There are a few cultural differences, but not too many.
In Slovak, the use of imperative is more acceptable, so people may sound bossy, while they are just being direct and open. Don't take this personally, if they sound rude, as it may be completely acceptable to communicate that way in our language.
If you ask "How are you?" be prepared to hear the whole story sometimes. However, it is okay not to ask, it is not considered a greeting in Slovak.
If you are gonna eat meals with other Slovaks, we say "Dobru chut!" (Bon apetit) before starting a meal together. Just plunging into your soup without saying it may look uncivilized.
Lunch is the dinner here, while suppers are light. So if you eat with other people, soup is just the starter and then a big meal comes right after that at lunchtime. However, in restaurants, you can always order a lighter meal or it is completely acceptable to skip the soup, even if the others have it.
In restaurants, "Menu" is the "Daily Special" and what is called "Menu" in the English language is called "A la carte".
In case anybody invites you to their home, it is generally considered polite to remove your shoes right past the front door. They would sometimes offer you visitor slippers. If not the case, they will ask you to keep your shoes on.
It is a custom (a rather silly one), to keep offering something to guests at least two or three times, as Slovaks tend to refuse every offer the first time to see if the one offering is serious. Be prepared to get the same thing offered to you three times, eg. a dessert, coffee. That's why the imperative is acceptable. My American missionary friend pointed out that sounding too polite in Slovak sometimes means that the hearer may think that you are not very serious. "Have some dessert!" may sound bossy to a North American, but it just means that this person really means that they want you to have it. I know, sometimes boggles my own mind :D. If you meet with people who have been to NA, they may be acquainted with your culture and it may be no issue.
There is quite a bit to see in the area, both history and nature-wise. The Kosiceans are very proud of their city. I mean very proud. However, some of my NA friends said that if you had some time to go travel, then Budapest, Vienna, Krakau and Prague are world class cities not too far really worth the visit.