The main airport for international arrivals is Kiev Borispil (KBP) although smaller European airlines fly to Zhulyany Airport (IEV). Cheap flights are available from within Europe with low cost airline carriers Wizz, Vueling, LOT & Pegasus all flying to Kiev. You can find cheap flights to Kiev on Skyscanner or (Travel Payout Russian site). UIA (Ukraine International Airlines) is the flag carrier and largest airline in Ukraine.
Borispil is located 30km from the city centre and there are many options to get to the city centre from the airport including Taxi, Train & Bus. Booking your transfers is essential and it will save you lots of money.
Kiev also has excellent rail links with most of the big cities in Europe including Paris, Amsterdam, Rome and London. Although you will most likely have to change in either Poland, Germany, Hungary, Czech Republic or Switzerland. Prices are affordable although it is worth remembering that they are not “high speed trains” and more often than not, it will be a night or sleeper train. Train tickets can be purchased on either Interail or PolRail.
If you’re doing Kiev, then you should consider Russia too. Although Visa’s could pose more hassle and expense, connecting the dots between both ex Soviet super powers. There are several direct night trains linking Kiev with Moscow, Prices vary depending on board but the average price is $XX and it takes around 12 hours. You can book tickets to and from Moscow via XX here.