There are 4 operators in Tajikistan: Babilon-M, Megafon, Tcell and Beeline. Coverage strength varies a lot. Babilon-M and Beeline come recommended in Dushanbe, but for instance in Murghab, Megafon and Tcell are better. A difficult choice for travellers.
Locals who travel a lot carry 2 or 3 sim cards. Roaming with your home country sim card should work (but probably very expensive), but funnily enough, roaming often does not work with a sim card from a neighbouring country.
In general, Megafon is a good choice.
Getting a sim card
The first places you can buy a sim card when entering Tajikistan from different border crossings are Murghab, Khorog, Isfara, Khujand and Dushanbe.
Sim cards are very cheap, but SIM cards are sold only for locals or foreigners who have registration. None of the providers give any guest/tourist tariffs. You have to show your passport when registering a sim card in Tajikistan and leave a digital fingerprint.
Or not? The rules and regulations about buying sim cards change very frequently and are not applied evenly across the country. Expect everything. We follow up on the situation in the Tajikistan sim card forum topic.
Mobile internet is not so expensive and if you can get a good signal, it should be fast enough to do research and stay in touch with people. 3G is available in the major towns. Outside, expect internet to be very slow.
Dushanbe has coffee places with wi-fi, your accommodation might have decent wi-fi too. There are internet cafes in places like Khujand, Penjikent and Khorog but not in Murghab. In Khujand, try Tajburger and Milano Pizza.
Right now, Megafon is recommended for Eastern Pamir, with T-Cell better in the Wakhan Valley. We have a forum topic with the latest on mobile internet on the Pamir Highway.
Internet in Tajikistan, like elsewhere, is filtered. You are being watched and unwanted websites are blocked. Social networks are sometimes blocked.