There are two main answers: Turkmen and of course, Russian, and it can be hard to know when to use each one. As a general rule everyone does speak Russian, but most people do appreciate the effort if you’re able to throw in some Turkmen, even just as a gesture. This is a common theme throughout Central Asia, though it has manifested slightly differently in each country.
The first official language of Turkmenistan is Turkmen, which is written in Latin letters (usually, although there are still some remnants of the modified Cyrillic script that was in use until 1992), and is a Turkic language, very similar to Turkish and Azeri. This is the language that is encouraged by the government and society in general, and to try and use a few basic words or phrases of Turkmen is a nod to their heritage and separate identity to that of their recent Soviet past.
Here are a few basic words that will get for sure get you a few smiles and might even come in handy:
Thankyou: Sagbol or Rakhmet
Beer: Piwo (taken from the Russian)
What is the English level in Turkmenistan?
You’ll actually probably be surprised at the level of English spoken in such a closed-off and un-international country. Especially in Ashgabat you’ll find plenty of people that speak at least a little bit of English. This is certainly not to say that it’s everyone, just that you might have been expecting it to be very few. It can be difficult to find opportunities to speak to locals, but occasionally you will come across someone who speaks English well. For the most part though, the people you’ll be interacting with are either staff at your accommodation who will speak reasonably good English, and servers at restaurants/cafes/bars, who will usually speak basically no English.
Read more about language in the region here.