Good call, having your own vehicle makes a huge difference once you want to get beyond the main cities and see the steppe properly. Public transport links the big spots, but the real magic is out on the open road and you simply cannot reach it on a bus. Bring your passport, a valid driving licence, and an IDP is handy. To browse options for a rental car in Kazakhstan just check the listings on this site
https://getrentacar.com/en-US/kazakhstan .