Ukraine in general has a reliable railway system. The faster trains between Lviv and Kyiv run at night and the journey lasts about 8 hours. Recently a new fast daytime train was started between Lviv and Kyiv. Crossing into Ukraine by train from the West would take longer as you have to go through customs and the process of changing train wheels to fit Ukraine's wider railway track gauge. You can shorten this delay by taking a train to Przemysl (Poland), where Ukraine's wide gauge tracks begin, and from there change for the Ukrainian train to reach Lviv, Kyiv and other destinations in Ukraine.
In Lviv you can buy railway tickets at the main railway station's ticket office, or at the ticket office in the city center at 20, Hnatiuka Street. Your ID is no longer required for buying train tickets in Ukraine. Be prepared to stand in line to get served.
Tickets for International destinations can be bought from Window #1 at Hnatiuka Street, or at Window #20 at the main train station. #20 is reached from Platform 1 following the sign that reads in Ukrainian "Зал підвищеного комфорту" where you may be charged an extra UAH 5 fee to buy tickets.
The most convenient connections could be made from Prague, Bratislava, Warsaw, Wroclaw, Krakow, Przemysl and Budapest. No online reservations are yet accepted by the Ukraine Railways but expected in the near future.