What's the best way to travel on a budget in Ukraine and Poland?

If you want to use the trains in this part of the world you better speak the languages. I live in Ukraine and would be VERY hard pressed to buy a ticket, read it, board, etc. without my girlfriend (who speaks Ukrainian). I can do well in Hungary because I speak Hungarian, but unlike airports, where all announcements are in the local language and English, all announcements in train stations are only in the local language. There is also a problem with crime on trains in Eastern Europe, especially since you're wanting to go to the "less-visited" parts of the country alone. In Ukraine, for example, you should buy a compartment with a door that locks, but that means purchasing tickets for at least one other bed (for which you will need another pport). You are safer to go with a tour group in this part of the world. Crime is not everywhere, but you need to be careful and not get yourself in a situation where you cannot deal with the language barrier. You can travel quite nicely for $500 for a week, but you have planned a little much I think for that week. Pick either Poland or Odessa and focus on that area. Find a reputable travel agency to advise you on tickets and purchase them for you. And make sure you don't get yourself in a place you can't get out of, just in case the worst happens.