Badges can be unlocked sometime after the campaign takes you out of Old Factory. If badges aren't unlocking for you, try progressing a bit more through the story.
Badges are earned by using a weapon, killing specific enemy classes, or using different traversal methods repetitively.
Grind Badges are earned by simply grinding on any rail or wire. In order for a grind to count towards it's badge, you must change rails. Grinding in a circle will not count towards the grind badge.
Here's a good method to "grind" this badge:
This challenge shouldn't be to hard to do regardless. Most travel done in the game is through grinding, so it should just unlock naturally.