Arvid Coltorp suggests maxing out the following skills first:
(P.S.: If you struggle with keeping your mood up because of skilling, read the Mood FAQ)
To be able to jam the songs in your repertoire to 100%, you need to learn the history of your band's genre. The book is called History of "genre".
You need to learn the basics and advanced books of the primary and secondary band role you picked. This is a large part of your personal rating on stage. Make sure that your band leader sets your instruments, otherwise they won't be taken into consideration for your concert review!
Note: The better charm you have, the more successful you will be
EDIT: According to Frank, this is now working: threadID13925.8
Note 1: You should be good health and good happiness when writing a song
Note 2: You should have amazing musical talents or be trying to raise it
Note 3: Songs will degrade in quality if you start working on more than one song every three days. The more songs you create over a three day period the less inspired they will be.
Note 4: The quality of the song is already set, when you start working on it. So if you improve your songwriting skills while working on a song, this won't influence its quality.
Note 5: Songquality can be further improved by seeds.
To not break your neck on stage, you need to learn some skills to perform the stage events well. Most important is Basic Showmanship and Basic Manners. You learn manners from your mom after finishing her tour, so make sure that you please all her wishes and you'll get 5* in it without having to skill it.
See the genre page to find out more about the rest of the required skills.
Also there are 5 skills, that improve your performance slightly and are also needed for some events.
Read the page on jobs to find out, which skills allow you to earn top salaries.
Extra:
The quality of books, that you already read, but accidentaly read again will turn truely abysmal.
You can improve skills beyond 5* by teaching them to apprentices.