Ryazan has a network of trams, trolleybuses, and buses connecting the various districts. The historic centre and Kremlin are walkable from each other. Taxis are plentiful and can be hailed or called through standard Russian apps. The city is compact enough that most visitors will not need much transport once in the centre.
For the excursion to Konstantinovo (the Yesenin birthplace), buses run from the Ryazan bus station, but a taxi or hired car is more convenient given the limited return service.