2 big operators remain in Uzbekistan: UzCell and Beeline. They are similar in terms of prices and coverage – a third operator, UMS is smaller.
Mobile internet is cheap. For 10$ you can get a sim card, call and go online for a month. Your western sim card will most likely work on roaming in Uzbekistan.
The main issue before was getting a sim card. Telephone companies would not register foreigners, so you needed a friend in Uzbekistan who wants to register your sim card in his name. In December 2016, this was not an issue anymore, but the problem could pop up again.
Internet cafes exist in the big cities, but don’t rely on them: bring your own device and access internet in your accommodation if it is available. Many internet cafes do not have internet (only games) or need your ID, especially in Ferghana Valley, Karakalpakstan and Termez.
In Uzbekistan, a lot of websites can only be accessed through VPN/proxy: news sites, blogging sites, torrent sites, opposition sites, porn sites. The main social networks remain accessible though.