Roaming works in Turkmenistan. You can call with most Western sim cards on roaming inside Turkmenistan. Funnily enough, not with an Azeri or Uzbek sim card. Check how much it will cost with your operator.
Internet in Turkmenistan, like elsewhere, is filtered. You are being watched and unwanted websites are blocked. Most social networks are also blocked. Good proxies and VPN’s on the Silk Road are Psiphon and HotSpotShield (other recommendations are welcome).
Very expensive roaming mobile internet on your home sim card works. A better way is to buy a Turkmen sim card. Speed is acceptable, prices are reasonable. All you need is your passport. MTS has an office at the Russian bazaar, but you can buy it elsewhere too.
There are a few good places for wifi in Ashgabat. 1 place with fast free wifi is the coffee bar at the Sofitel Ashgabat Oguzkent Hotel. Even better is the Public Affairs section of the US Embassy (3rd floor, Business Center, Andalyp street, 70) Open Mo-Fri 9-13 and 14-20, Sat 09-12 and 13-18.
Everywhere else, it will be highly frustrating. A few high-end hotels in Ashgabat, Balkanabat, Mary and Turkmenbashi have internet, but it’s very expensive (5$/hour is normal), very slow and unreliable.