There are two International airports serving Kyiv.

The main airport is Boryspil International Airport located 29 km east of Kyiv. The main terminal D is new and the services are good. The passport control and custom checks are well organized with short waiting lines. The collection of suitcases is a smooth process. 

Often expats are picked up by their organization upon their first arrival. If that is not the case then the best option is to go by taxi since there is no metro/train (and the bus is rather complicated to figure out). At the exit there is a desk that provides taxi services. It is around 350 hrivna ($15) for a 45 minute ride to the city center. Once you are more familiar with the taxi services the best way (cheaper) will be to use their call-in service.

When using a taxi it is recommended to have the address written in Cyrillic though most taxi drivers manage to recognize the street name when one would pronounce it in English. You can ask the airport taxi service to assist in writing the address in Cyrillic.

The smaller airport is Kyiv International Airport in Zhuliany offering the budget airlines. Terminal A is the newer and serves international flights. 

Zhuliany airport is located about 7 km south of the city centre. The best options to reach the downtown is to get a taxi. But also various buses depart from outside the airport (http://iev.aero/en/airport/contacts/).