I know a few rules
- number ends with even digit, it is divisible by 2
- number ends with 5 or 0 is divisible by 5
- if some of all digits in a number is divisible by 3 then that number is divisible by 3
How many more rules are there for division?
All of these ad-hoc divisibility tests are absolutely trivial if you know congruence / modular arithmetic. For example, see my discussion of casting out $91$'s linked in my answer to a similar prior question.
Therefore, if you desire to obtain some universal insight into divisibility tests, I strongly recommend that you first learn modular arithmetic. It's a protypical example of a ubiquitous algebraic problem solving technique - modular reduction. Due to this generality, the time invested in this endeavor will provide great rewards in your future studies.