From the different operators in Kyrgyzstan, Megacom has good reach, low prices, the best speed and good customer support. Beeline comes in second place. Roaming should work with your foreign sim card, but generally not with sim cards of neighbouring countries, funnily enough. Opensignal has a map of the reach of each network.
Mobile internet is inexpensive and fast enough to do research and stay in touch with people. 3G is available in the major towns. In Central Asia you need to always discount 1G compared to the Global North, though. 4G = 3G and 3G = 2G. Bishkek has plenty of coffee places with wi-fi and your accommodation should have decent wi-fi too. Internet cafes are easily found in the big cities, but small towns usually have an internet cafe as well. This will be a dark hole filled with gaming teenage boys.
There are few restrictions on the internet in Kyrgyzstan. Only a few political websites are blocked and the government does not have the resources for a comprehensive internet filter. You are only being watched by Silicon Valley and the US government. Good proxies and VPN’s on the Silk Road are Psiphon and HotSpotShield (other recommendations are welcome).