Is there any systematic way to look for finite non-abelian simple groups? or any clever idea which makes them easier to find?

Question

I know that any finite abelian simple group like $G$ have the following properties:

1. $|G| = p$ where $p$ is a prime number.
2. $G \simeq \mathbb{Z}/p\mathbb{Z}$ (more general, we can say any simple abelian group is a cyclic group of prime order)

So we have a systematic way to find finite abelian simple groups, it's equivalent to find $\mathbb{Z}/p\mathbb{Z}$ where $p$ is a prime number.

What about finite non-abelian simple groups (like $A_5$ and $PSL(2,7)$), is there any systematic way to find them? or any clever idea which makes them easier to find?