Your questions is rather broad, as we can have all types of patterns that can include logic patterns, number patterns, and even word patterns.
For number patterns, there are all sorts of things (list not exhaustive) to learn, investigate and explore, such as:
$\bullet$ Arithmetic Sequences
$\bullet$ Geometric Sequences
See for example, the Number Sequence Calculator
$\bullet$ Special Sequences
$\bullet$ Square Numbers, Cube Numbers, Fibonacci Numbers, Pascal's Triangle, $\cdots$
$\bullet$ General Number Patterns
$\bullet$ Repeating Patterns
$\bullet$ Method of Common Differences
$\bullet$ Non-Math Sequences
$\bullet$ Mathematical Series
$\bullet$ Discrete Mathematics
There are also very interesting patterns and here are some images of those.
There are lots of general techniques like taking the first difference of members (works for quadratic equations, linear recurrence relations, etc.) or looking at subsequences (say, every other member).
You can browse through the OEIS and see the many types of relationships you can have (from the very simple to the very complex).
This site has over 400 examples of Number Patterns and look at the magic anything graphics and patterns.
There are entire books written for all kinds of patterns and symmetry within mathematics.