Although the city has a railway station, I always take a bus to get there from Luhansk.
All the tourists coming to the city to honor the war heroes arrive to the bus terminal.
The railway station remains in the use of the locals, so to say. It is in the neighboring town of Sukhodolsk, a satellite town of Krasnodon.
There are buses to Krasnodon leaving from Luhansk bus terminal almost every half an hour. They will charge you around $3 to take you to Krasnodon within an hour.
The cab fare to Krasnodon is about $40.
There are a lot of cab services in Kranodon.
Here is a business card of one of them.
You can hail a cab almost at any bus stop or at any restaurant or supermarket.