As someone who has spent far too much time thinking about classifying groups of small order, one thing you'd want to do is understand some more results about p-groups. You already probably learned one of the main results, every p-group has a nontrivial center, which is proved by the number theory type arguments you discussed. One of the key ideas in understanding p-groups is the Burnside basis theorem that says that any generating set for $G/\Phi(G)$ is a generating set for G. Here $\Phi(G)$ is the Frattini subgroup which is generated by all commutators and all pth powers. You can think of this as a version of Nakayma's lemma for p-groups and think that p-groups are similar to commutative rings.