The biggest benefit of a magic doctrine is that it unlocks new spells while providing some inherent effects (just like a civic).
First, you have to research the magic doctrine which will become available in the 
Governing Adviser. While you can cast spells form inactive doctrines, they will cost a small amount of 
Premium Points* in addition to the spell 
energy cost. If you find yourself casting often spells from a certain doctrine, switch to that doctrine to save Premium Points.
* Premium Membership removes the Premium Point cost.