Those few advices might save you a lot of time and head ache, they are very easy to remember and are not difficult to follow!
Carry the passport alongFirst thing you should remember: at any time when you are out of the place where you stay you need to have a valid passport with you. Ukrainians who you know might advice you otherwise, most of the locals don't bother to carry their IDs along with them, but you are a foreigner and it is much easier just to keep your passport close than explaining the police why you don't have any document to prove your identity.
According to the current laws of Ukraine the police personnel is allowed to check ID of any person who might look or act suspicious (in reality, they can stop any person who they want to stop and you don't need to do anything funny to be stopped!). It's very important that you have your passport with you. I was asked several times if a copy of the passport or driving license can be used instead of the original passport. The answer to this is in the "grey zone" it might work sometimes, but the copy of the passport can be forged easily, so the police can refuse it, and the driving license issued by the US authorities doesn't necessarily replace passport.
Learn a few phrases in RussianYou need to try to learn few phrases in Russian, so that you can explain that you are: "American tourist- amerikanski toorist" and that "you don't speak Russian well ya ne govoru pa Rooskee!". Try to make sure that you passport actually stays in your hands when you show it to the police.
Ukraine visa regulationsIn case you didn't need a visa to enter Ukraine, you can stay in the country for 90 days since the day of you arrival. If you are a citizen of the USA your passport will be stamped on the border of Ukraine when you enter its territory, this stamp is the proof of the fact that your stay in Ukraine is legal.
No alcohol in public in UkraineDrinking alcoholic beverages in public is prohibited in Ukraine, this law is also enforced by the police, so never drink beer in the public parks. You can order and drink alcohol in the establishments located in the same park, but not outside of them. Breaking this rule puts you in a trouble and it doesn't matter if youhave your passport or not. The same thing with drinking and smoking in the trains, there are fines for that.
Enjoy your stay in UkraineUkraine may sometimes seem to have too many rules and regulations, but in fact visitors of this country very rarely face any problems. I have been working for the Mordinson Ukrainian marriage agency for more than 10 years and we've had hundreds of clients coming over and staying in Kharkov and I can say that there were never any serious problems.
Enjoy Ukraine! Sincerely, Michael Mordinson.