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. A proxy, VPN or SSH tunnel is a useful tool.
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 low. All you need is your passport, registration should be quick.
Your updates and questions regarding mobile connectivity in Turkmenistan are welcome in this forum topic.
There are a few good places for wifi in Ashgabat. Fast free wifi is available at the coffee bar in the Sofitel Ashgabat Oguzkent Hotel, or get a coffee at the Grand Turkmen 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.