Difficulty in proving the existence of one-way functions
Apparently proving the existence of these functions would be more difficult than proving $P\not =NP$? And I'd like to know the technical reasons for this, does it have something to do with the "for every" in the definition?


The answer is in your question itself:


  I believe that since proving its existence would also prove $P\not =NP$


This is indeed the case. Somebody that proves the existence of one-way functions automatically also proves $P\not = NP$. On the other hand, it is conceivable that you could prove $P\not = NP$ without proving that one way functions exist. Thus, proving the existence of one-way functions is at least as hard as proving $P\not = NP$.


  What steps would a mathematician or computer scientist take to prove that one-way functions exist?


Coming up with an extremely clever and creative idea that nobody came up with before.