Several private companies operate a daily bus service between Moscow and Elista. The pickup point in Moscow is Luzhniki Stadium. A one-way ticket costs about 1350RUR.
By plane
By plane
Flights from Moscow are a few times a week on Rusline. There are more regular flights to Volgograd and Rostov, with transfer to Elista by bus or train respectively.
By train
By train
The Moscow-Elista train route goes through Stavropol. However, the most convenient way is travel to Volgograd by train then on to Elista by express minivan, which are relatively cheap.