Sorry, one more question: I know that to improve interpersonal insight we need to visit the Guest Hall, but how do we improve the basic insight ?
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It can be raised several ways: picking specific dialogue options, receiving gifts from others, and under your Free time to-do list - select Ask someone for skill advice - and then select Practice dancing with Jasper.
Most other skills can be raised through exploring the castle, asking someone for skill advice, or self improvement.
"Insight" kind of translates to "meta knowledge about the game." It's basically a measure of whether or not you've played the game before and how well you understand the characters. During the welcome feast at the beginning of the game, you have the opportunity to gain a total of 70 insight. To do that, you need to show up early so you have time to talk to as many characters as possible.
There are guides out there explaining how to get the max amount of insight on day 1, but it's possible to figure it out yourself. Each character you talk to asks multiple questions. They typically ask you 1 or 2 questions that require a skill check and 1 or 2 questions that do not. For example: when you meet Cordelia, the first question is just how to greet her. The greeting is based on skill checks; you can greet her politely if you have enough etiquette, curiously if you have enough intelligence, etc. Her second question is another skill check. Her last question is not a skill check. This is the question that grants insight. She politely asks you how your journey was. The correct answer is the most polite one. It makes her happy and grants +5 insight. All the conversations during the welcome feast share the same basic format. The questions that do not require a skill check, the ones that are just about choosing what will make that character happy, those are the ones that grant insight.
The welcome feast will grant you the most insight, but there are other opportunities to raise insight scattered throughout the game. They typically come up during group events. For example, during the introduction speeches at the end of week 1, you can get +5 insight from answering Woodly's question. And then during the horse ride, you can get +5 insight from answering Yvette's question.