How to Level Up Zombies
Zombies Earn Exp With Tasks

Once you've unlocked Zombie management which is done by helping the Plague Doctor, you can start assigning them to tasks around your workshop. Different tasks provide different talent exp to a zombie; i.e., stonemasonry gives red exp, gathering lumber gives green exp, etc.
Assign Zombies Task Appropriate to their Skill Level

A Zombie's mastery level must match or be greater than the task they're assigned to. If not, the Zombie cannot perform the task and won't earn exp.
How to Increase Zombie Mastery
To make a Zombie reach the necessary talent mastery to perform tasks, you can either:Equip Zombies With Them Tools
Equipping the right tools on a Zombie can give them a flat mastery boost to a specific talent. Keep in mind that removing these tools will remove their mastery boost and is non-permanent.
Increase Zombie Mastery Via Perks
With the right amount and type of exp, you can manually increase the base mastery level of a Zombie. This takes longer, but the mastery level is permanent.
