The official language of Turkmenistan is Turkmen, part of the Ural-Altaic language group which includes the family of Turkic languages. Within this group, Turkmen is one of the Oguz or Southern Turkish languages, along with Turkish and Azeri. There are many Turkmen dialects, broadly corresponding to the different Turkmen tribes. The standard Turkmen language is based most strongly around the Yomud and Tekke dialects. During the Soviet period, the Turkmen language adopted many Russian words, especially to describe new items of technology. The Turkmen government is encouraging their replacement by equivalent terms derived from authentic Turkmen roots.
Turkmen has been written in a range of alphabets, reflecting changing political priorities. It was written in an Arabic script until the late 1920s, when the Latin-based Unified Turkish Latin Alphabet was introduced, similar to the alphabet currently in use for the Turkish language. A Cyrillic script was introduced in 1940. Following independence, President Niyazov introduced a switch to another Latin alphabet, known as the New Alphabet (Taze Elipbiy). The president even patented the script.
Russian-language teaching was given a high priority in the Soviet period. Better-off families in urban areas, members of the Russian and other non-Turkmen minority groups and many older Turkmens speak Russian, and so a knowledge of this language is invaluable for getting around and making conversation in Turkmenistan. The Russian-language ability of the young generation is, however, markedly weaker than that of their parents. While Russian is still taught in many schools, it is as a foreign language rather than a medium of wider instruction. In rural communities you may struggle to communicate in Russian.
English is also taught as a foreign language in some schools, but the standard of tuition is patchy at best. You will certainly find many Turkmens eager to practise their English, but this often extends to little more than a few standard phrases.