Most of Europe uses sim cards for phones. Ukraine has has a couple carriers Kyivstar GSM, MTC (former UMC), Djuice, Jeans, Beeline, Life :), U'TEL, Mobilych, UNI.
AT&T and T-MOBILE use the same technology and will unlock your phone for free if you meet certain requirements. Once your phone is unlocked you can just buy a sim card that fits and you can make cheep local calls. Ebay and Amazon also sell unlocked quad band phones for use over seas. For a small fee you can use internet on your smart phone. MTC charges .01 UAH for a month of unlimited ( rates subject to change). There are a few places at the airport to purchase a sim and recharge, there are also kiosk all over the city and many small shops offer sim and recharge. There are also automated machines to purchase more time and internet service. Data for smart phones is really cheap compared to other country's. Check with the carrier about terms and fees for usage, most offer unlimited for a very reasonable fee.