Each of the magics you encounter will have three levels of mastery with each one effectively granting a single magical ability.
There are two systemic requirements for gaining each level of a magic are:
Persuade a suitable teacher to share their secrets with you. Mages are almost universally jealous of their secrets so you'll need to charm, threaten or blackmail them into helping you. Also Master mages are extraordinarily rare, so finding them may be an extremely difficult challenge in its own right.
There are many different magics in the world and you may not be suitable for them all. If you aren't suitable to be a student for a particular type of magic then a teacher will tell you so upon your first action, and tell you a bit about the magic so at least you will have some greater knowledge of the subject.
Make an appropriate sacrifice and learn the secrets of that level of magic.
Sacrifices vary between magics but the severity is always as follows:
You need to put your magic to use for at least a turn before you can Sacrifice again, but you can put the effort into improving relations with your teacher/finding a new tutor at any time.
So an example of learning magic at the maximum speed. Here we use the traditional example of Custardmancy. Each bullet point is a major action.