Hi, dexterdev. There are two nice proofs of what you're talking about at Finite field - Wikipedia, the free encyclopedia in the section "Detailed proof of the classification." If you're trying to avoid things like Z/pZ and vector space, maybe you'll find the second argument more appealing. For the second proof you need to be familiar with Cauchy's Theorem: it says that if A is a finite group and p is a prime divisor of the order of A, then A has an element of order p.
Don't be afraid of the notations and things you don't know in algebra. If you really want to understand, you look it over enough and are willing to ask questions, you will get it.
Take a look at the arguments and feel free to ask any follow up questions. Good luck!