Could someone please explain how the skill leveling works?

There is LUK (drop chance), and two bars to the right named "Special Skill 1 (Level 1) chance" and "Luck 11 - Special Skill 1&2 level".

How do they correlate with upgrade materials, senki level and LUK? The guide is not clear and translation errors might give the wrong idea.

Is it OK to simply level up the senkis without caring about skill levels? Or is it imperative to somehow increase those luck bars when the senki is low level?