In Ukraine, everywhere there are plenty of ATM (cash points), to get money from your credit or debit cards. You can use everywhere cards with logos
Visa (Visa Electron, Visa Classic, Visa Gold etc.) and
MasterCard (including EuroCard, Maestro, MasterCard Electronic etc.)
These types of cards you can use to pay in supermarkets, some of petrol stations, most of hotels, restaurants.
Cards with logo American Express is accepted only in very limited number of hotels or restaurants. None of supermarkets, none of ATM, most of banks do not accept this type of card.
Most of banks in Ukraine accept Travelers cheques. Inconvenience, is that you can cash your cheque only on workdays and only in bank, as most of banks are closed on week ends. (in contrast to credit cards, when you can get a cash at any day any time in one of plenty of ATM everywhere.
What ever you pay in Ukraine, you should use Ukrainian currency - Hrivna. Cash (US dollars & Euros) you can easily exchange into hrivna in one of plenty Currency Exchange Offices. You can see such offices everywhere, in airports, train stations, halls of hotels, on street, near markets etc. Of course, exchange rate is always against you, if you change money in hotel, airport or train station.