Just got back from a trip to Kyiv via Warsaw, Przemysl and Lviv.
The Hyundai-Rotem inter-city EMUs run on several routes radiating from Kyiv (not to occupied Donetsk/Luhansk obviously).
One place I noticed with several services was Vinnytsya SW of Kyiv, which is about a 3 hour trip.
It's on the route of the Odessa and one of the Lviv services, so runs several times a day.
I did Przemysl-Lviv on one, but the buffet was closed and only took Ukrainian currency anyway (which I didn't have at the time).
The UZ site can sell you e-tickets for long distance trains:
https://booking.uz.gov.ua/en/
Most of the long distance trains are loco-hauled sleepers, though, converted to day use.
For 2nd class this gives you a rather narrow compartment with bench seats for 6 (and possibly someone sleeping in the top bunk!).
The "Borispil Airport" train is a 2-car DMU running hourly from P14 at Kyiv, fare 80 UAH (£3) for 30km.
The airport branch is a 2km unwired single track off the Kharkiv main line in the SE of the city.
You haven't said how you are getting to Kyiv, but if by air, the low-cost guys (Ryanair etc) use Terminal D, which is a shuttle bus-ride from the main terminal F used by BA and the main airlines (and where the train is). Bus every 30 minutes.
I used terminal D to depart, very easy really, but note there are no shops or duty free there - just a couple of coffee outlets.
The metro in Kyiv is impressive, very deep stations with faster escalators than we are used to.
Fare for every trip metro/tram/bus is 8 UAH (30p) with a very easy smartcard system for multiple journeys.
In the few hours I had I visited several orthodox cathedrals with golden domes, plus the Golden Gate and an excellent performance of Carmen at the Opera.
Thought about paying my respects at Babi Yar (WW2 massacre site) in the NW of the city, but ran out of time.
Kyiv/Ukraine is indeed a fascinating place, rail and otherwise.
I can also say it felt a good deal safer late at night than many western European cities including some of our own.
You can pay for almost anything by card, but I never found an ATM which coughed up more then 200 UAH in cash (about £6).
One good planning site is
https://www.eway.in.ua/en/cities/kyiv
It's a bit like flightradar, but for public transport.
It shows all the routes and where all the vehicles are.
Apart from metro/trams etc there is also a circular heavy rail route around the city which should also cost just the 8 UAH.