If you are interested in this type of mathematics, you could do worse than to read through that question and the answers to it. If you want to know about general lower bound results, the Westzynthius paper has a nice construction which will produce gaps between primes which are larger than average, also using elementary means; it was the first published construction to show that for any constant C there are infinitely many k such that $p_{k+1} \gt C \log p_k + p_k$. You might even find a way to make elementary improvements on the arguments, as well as search the literature to find improvements by Rankin, Erdos, and others. (If you are patient, you can wait for a writeup I am doing which includes an interpretation of the key results of the paper.)