As for eating out, unless you can speak the language, be careful. I tried borsht in one restaurant only to have fish in my mouth for the first time in twenty years. There are Ukrainian dishes you can try, however. Deruny are spicy potato pancakes. I do not eat meat or dairy products.
Many Ukrainian dishes are heavy with meat and dairy products.
I try to let my hosts know about my diet ahead of time, preferably in casual conversation. If they don't get it quite right, I relax my rules a little to include foods made with dairy or egg ingredients so as not to be rude. Remember, this is unusual for them. And I always eat or drink something. Imagine how they would feel since they often have so little, if you refuse to accept what they offer.