There are no trails in Kazakhstan, besides those that circulate through word-of-mouth in the local climbing community. There are also no updated topographic maps in Kazakhstan; we are still using the Soviet military maps.
So for the beginning hiker , we recommend going on a guided tour or downloading a gps trail, a place like Wikiloc already has some nice gps-hikes on offer. For the advanced bushwacker: make your own trail!
There are still many places to discover for us, we only just started. A few day hikes close to Almaty are up already, everything else is still to-do. On the map below, find the best areas for hiking right now AFAIK. There are more prime places, like Altai, the far southeast, the Dzhungar mountains, … but issues with permits and the lack of knowledge I have about them means I did not put them on the map yet.