The historic centre of Kazan — from the Kremlin south to Tukay Square, taking in the Old Tatar Quarter and Bauman Street — is small enough to walk in any direction in twenty minutes. Walking is the easiest way to see it.
The Kazan Metro is short (a single line, eleven stations) but cheap and useful for getting to the railway station, the central market, and the further suburbs. Buses, trolleybuses, and trams cover everywhere else; pay with cash or a tap card. Taxis are abundant and cheap — use the Yandex Go app (available in English) rather than hailing on the street. Driving in the centre is best avoided because of one-way systems and parking; rent a car only if heading further afield to Raifa or Sviyazhsk.