There are 3 operators in Kazakhstan: KCell, Beeline and Tele2. Tele2 is the cheapest, but has the least reach. If you are not planning to get out of the city however, this will do just fine. Sim cards are free or almost free. Roaming with your western sim card should work, but funnily enough, not with a sim card from a neighbouring country. Opensignal has a map of the reach of each network.
Hotels, hostels and coffee places in the big cities have wifi. Otherwise, tracking down an internet cafe should be easy. For mobile internet, once again, Tele2 is the cheapest, Beeline and KCell have better coverage. It is cheap in general. 3G is everywhere and 4G has arrived with Altel, only in Almaty and Astana. In Central Asia you need to always discount 1G compared to the Global North, though. 4G = 3G and 3G = 2G.
Internet in Kazakhstan, like elsewhere, is filtered. You are being watched and unwanted websites are blocked. Social networks are not blocked. Good proxies and VPN’s on the Silk Road are Psiphon and HotSpotShield (other recommendations are welcome).