In Ukraine there are 3 major mobile operators: MTS, KyivStar and Life:). You don't need to show your passport to buy a mobile operator card. When you use a card of one of Ukrainian mobile operators, you don't pay for incoming calls. Outcoming calls are rather cheap (including calls abroad). All mobile operators have tariff plans with free calls inside the network of this operator. It can be handy for group of friends or family, traveling in Ukraine. Cost of card is very low.
Each Ukrainian mobile operator has also mobile Internet offers. That's if you travel with your laptop, you can connect to Internet using your mobile phone as a modem. You can connect your mobile phone via USB cable. In this case you should install your phone as a modem. Or you can connect your phone via Bluetooth link (it will be the simplest). Tariffs for mobile Internet are very cheap in Ukraine. Each operator offers different tariff plans. Detailed instructions you will find on mentioned above websites.