Base mechanics:
All your skills (except knowledge skills) are capped at 50 until you have 25 in every skill in a category and unlock that category's outfit. Wearing that outfit gives a +10 to every skill in the correspoding category
The Empathy skill lets you understand how people feel about you. Higher Empathy rating means more specific detail.
Knowledge skills give their own bonuses. Science increases the speed of the ship at 50, Bureaucracy gives an additional social phase action at... either 50 or 75, unsure. Probably more of these that I haven't seen yet. Gaining this skill from your advisor will still provide this bonus.
A cabinet member will usually give you a bonus to learning their associated knowledge skilll, but can stop giving that bonus or even impose a penalty if your knowledge exceeds theirs.
General advice:
First impressions are important
Low approval with an elector means more drastic measures are required to win their support
High approval with an elector doesn't mean absolute safety. They can still pledge their vote to a rival!