There are 3 big 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 home sim card should work.
Wifi is ubiquitous in Kazakhstan, but sadly, these days you often need a Kazakh phone number to sign in.
Internet in Kazakhstan, like elsewhere, is filtered. You are being watched and unwanted websites are blocked. Social networks are sometimes blocked, and network speed is throttled when the government senses dissent. Sometimes, the government turns off the mobile network and internet completely.
A VPN is useful.
A law requiring mobile phone registration came into force on 1st January 2019. So far, it seems everything is working normally without the need for registration. Updates welcome in the forum.
Also in 2019, a rumour emerged that Kazakhstan would be able to intercept all https traffic. So far, this has not borne out to be true either.