Look out for a range of 'Turkmenbashi' labeled vodka, which can be washed down with the range of Russian 'Baltika' brand beer. It can be harder to find local beers in outlets catering to foreigners, but 'Berk' is well worth asking for; 'Zip', on the other hand, is awful.
Tea is excellent and readily available.
Best to err on the side of caution, and stick with bottled water. As in Russia, you may want to specify byehz gah-zah (literally, 'without gas' or 'still; plain') if you do not like fizzy water. 'Borjomi' mineral water from Georgia is available in Ashgabat's shops.
Local people prefer to drink gok chai - green tea, often with dried fruits or herbs, as mint.