General:
* Acquire
* Absorb
* Grasp
* Master
* Pick up
* Study
* Understand
* Comprehend
* Discover
* Figure out
* Get a handle on
* Realize
* Recognize
Formal:
* Assimilate
* Cognize
* Elicit
* Perceive
* Seize
* Fathom
* Deduce
Informal:
* Get the hang of
* Catch on
* Figure it out
* Soak up
* Get with the program
Specific to particular contexts:
* Memorize: to commit something to memory
* Educate: to teach someone something
* Train: to teach someone a skill
* Instruct: to teach someone how to do something
* Cultivate: to develop a skill or habit
* Develop: to improve a skill or ability
* Enhance: to increase knowledge or skill
The best synonym for "learn" will depend on the specific context.