I am very happy to reply to this thread because I visited Ukraine last year and it is now one of my favorite cities in Europe.
Please visit Kiev! This is essential. It is a big city in Ukraine, but for Americans who are used to capitals like Chicago, New York, etc., it feels small, so you don't have the worry about rushing around and doing everything before your time runs out. Things to do in Kiev:
- go to all the churchs - St. Sophia, St. Michael, St. Andrei, St. Volodymyr (my favorite).
see the botanical gardens if you go in the summer/spring. There is a small one and a large one - even if you don't generally like gardens, these are a must see - so beautiful.
go on Andreevsky Spusk - lots of nice souvenirs and fun atmosphere
see an opera - not expensive at all and good quality music
eat the food. Ukraine is very cheap and the food is number one in all the countries I've visited. I loved this chain restuarant called "Puzata Hata" which is cafeteria style and $4 can get you a very tasty meal, plus lets you try out all the native food.
I also went to Odessa. Kiev to Odessa is one night on the train and it's not far (9 hours?). (Kharkov is close to Kiev, but in opposite directions).
In Odessa, I stayed at a great hostel called I think "The Front Page Hostel" or something to do with newspapers. It was right downtown, super-friendly owners (one a Brit, one American), very clean and I meet a lot of very nice travels. I HIGHLY recommend this hostel.
What to do there - well, in Odessa, basically see everything that the guidebooks say. There's not a lot to do tourist-wise, but the beaches are very nice. There is the Pushkin Museum, Ptomekin steps, Deribasavski Street. There is a huge food market there - the oldest in Ukraine I think. It's definitely worth a visit.
Personally, I would just take the train everywhere, since it's cheap and if you take overnight trains you don't even notice hte time.