More formal:
* Discover: Implies a sense of finding something new or unexpected.
* Learn: Emphasizes the acquisition of knowledge.
* Ascertain: Indicates a process of finding out for certain.
* Realize: Focuses on the moment of understanding.
* Perceive: Highlights the ability to understand or recognize something.
* Comprehend: Emphasizes full understanding.
More informal:
* Figure out: Suggests a process of solving a problem.
* Find out: Indicates a search for information.
* Get to know: Implies a gradual process of becoming familiar with someone or something.
* Understand: Emphasizes the ability to grasp the meaning.
* See: Can be used in a metaphorical sense to mean "understand."
Other options:
* Become acquainted with: Focuses on the beginning of a relationship.
* Be familiar with: Indicates prior knowledge.
* Gain insight into: Implies a deeper level of understanding.
* Become aware of: Emphasizes the act of noticing something.
The best word to use will depend on the specific situation.