Virtually all large cities and towns offer some form of public transportation, for example buses and taxis which means going around in Ukraine is not a problem.
Info about almost all kinds of public transport in Ukraine which includes airlines, airports in Ukraine, bus routes, driving maps and trains in Ukraine are available at this website.
The tables underneath will cover additional information regarding the travel methods to get around Ukraine: Travel by rail, travel by air, and travel by bus and car.
Airports in Ukraine
Public Transport in Ukraine
Ukrainian Public Transport Questions