Buses and Marshrutkas (mini-buses) leave from just outside the airport whenever they get full. They cost 20 UAH ($5-6) to metro station Kharkivska or 25 UAH ($7-8) to the train station (vokzal) downtown. They are convenient, easy and you don't hardly need to know any English to use them. Just get in, pay and sit down.

Taxis are far more expensive, but they'll take you where you want to go, obviously.

No train or metro to airport, unfortunately.

Kiev is a wonderful city. I was just there a couple weeks ago. Have a great time!