What are the best routes for management skills growth?

Practice.  You don't need to have a the job title "manager" to use management skills. 

Volunteer to do things at work: little projects, awkward jobs.

Volunteer in the voluntary sector in a role that has management aspects.

Observe how managers around you behave, and the impact they have.  What is good?  Bad?  Why?

Think.

Read.  Start with Machiavelli's "The Prince" and von Clausewitz "On War".