Badges are acquired from Danger zones at a certain chance with the following detail:

Bronze Badge: Acquired as rewards in all Danger zones
Silver Badge: Acquired as rewards in Danger zones from Open Field to areas farther away
Gold Badge: Acquired as rewards in Danger zones from Badlands to areas farther away

Badges can also be acquired as quest rewards from annihilating zombies, as wells as by purchase with Bottle Caps in the Shop.

They can also be acquired in small quantity when exiling Survivors.