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 basic and advanced skills 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!

• Electric Guitarist, Acoustic Guitarist, Bass Guitarist, Violinist, Banjo Player, Chapman Stick Player→ Basic String Instruments → Play the Violin/Bass guitar/Banjo/guitar/Chapman Stick

• Lead Singer, Backup Vocalist, Rapper → Basic Singing → Lead Vocals/Back up vocals/Rapping

• Keyboard Player, Pianist → Basic Keyboard Instruments → Play the piano

• Drummer → Basic Percussion Instruments → Play the drums

• Trumpet Player → Basic Brass Instruments → Play the trumpet

• Dancer → Basic Dancing → Professional Dancing

• Flautist, Harmonica Player,Saxophonist → Basic Woodwind Instruments → Play the Flute/Harmonica/Saxophone

* Turntable Artist → Basic Electronic Instruments → Turntable Techniques

Note: This isn't a complete list of band roles you can choose. For all band roles go here.

• Basic Media Manipulation - Helps with talking to the press, which is most effective, when done by the Band Leader

Note: The better charm you have, the more successful you will be

EDIT: According to Frank, this is now working: threadID13925.8

• Lyrics & Poetry
• Composing
• advanced composing skills

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 are 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.

• Basic Fashion
• Basic Make Up
• Basic Sex Appeal
• Basic Showmanship
• Basic Dancing

• Basic Record Production - One person in the band should learn this, so your recordings turn out well. If you want to save this up for later, you can ask someone else to sit in the studio with you.

Read the page on jobs to find out, which skills allow you to earn top salaries.

• Health: To make your workout more efficient, even if you're not in the gym, you can skill up to 5* in either Basic Yoga or Basic Martial Arts.

There are rumors that both at 5* make the gain even better, but help files suggest that this is not true. You can further improve your gain by learning and skilling up either Yoga Mastery, if you went for Basic Yoga (Note:it's an underground skill,you can only learn this one through a teacher) or Jujutsu Mastery, if you went for Basic Martial Arts.

Note: We do not know if all four skills accumulate.

To improve your health with resting above 80%, you can learn Basic Medicine

• Mood: To be able to increase your mood during resting, you can learn Basic Religion

• Read the Mood and Health FAQ, to learn more about managing Mood and Health.

Extra:

The quality of books that teach the skill that you already have, but you accidentaly buy another of a same kind and read it will turn truly abysmal.

You can improve skills beyond 5* (up to 10*) by improving them on your own (right now appears to be up to 9,55% stars), improving then at a university and teaching them to apprentices