Etiquette and Empathy are universally useful. If nothing else, throwing a big party for your friends to efficiently spend two early days on these skills makes your life a lot easier.
Science (50+) provides an extra unit of movement during navigation. This is one of the most important bonuses you can obtain.
Bureaucracy (50+) provides an extra social action.
Media (70 sufficient, 30 is not) can let you poll the entire empire to see how every elector in zones you’ve ever visited feels about you. This also lets you see if you’ve upset any of your cabinet, which is a very bad situation. Character icons will always display the most recent source of information, even if that means replacing a precise “92% positive” assessment you had just performed in person with a “generally positive” assessment.
Law and Economics don’t seem to have an inherent bonus, but are relevant to many checks.
Advisors are important. The skill value used is your personal knowledge or your advisor’s knowledge, whichever is greater. Two Practiced ratings in the same subject will give you the Practiced rating, not sum to give you a Grandmaster rating.
There are a few specific situations where an advisor wouldn’t be able to help you, and you have to rely on your own knowledge. The earliest case is made explicit, where if you want to learn more about Korundi, you must be skilled enough to investigate on your own. Other cases can be less obvious.
Some events happen in response to you personally learning more about the subject.
Advisors can help you learn at +15 instead of +10. (Not in the actual lesson-as-social-action way, that’s limited to Kxenia and Vasily. I was surprised when Rejt wasn’t offering to talk about music on the side!) This bonus is based on how far away you are from their level at the start of the day. If you have an advisor at 50, learn half-day then full-day, you’ll end up at 45. If do it the other way, full-day then half-day, you’ll end up at 40.
Advisors can hold you back. The start-of-day logic is the same as for bonuses. The highest score you can reach with an advisor at 50 and six lessons is 75: half-day, full-day, half-day, full-day. If you would be learning a knowledge skill at +5 per lesson, you can fire the advisor to return to the basic +10 per lesson.
Firing a cabinet member usually makes them very unhappy. Depending on the relationship, you might be able to reassure them that it’s what’s best for the cause, or right now you need to keep your enemies closer. Some were never interested in keeping a position long term, but it’s very obvious.
You leave impressions on more than just Electors.
Moods can have many unexpected effects on events and interactions. A specific mood could open up possibilities that weren’t available otherwise, or even have you do something which is completely different from what you normally do. You will get a notification if something changed because you were affected by a mood. You won’t get a notification if something could have been affected by a mood.