Because of this should I compete in the routes by pushing lower prices or should I just stick with routes that arn't being serviced by the 2 major airlines.
I'm sure others will have a different strategy, but mine personally would be to do the following, especially in a very crowded world:
Initially, try and avoid competing directly with large carriers. As much as you can try to undercut their prices, they can do the exact same to you, but with one large difference: the large airline can run one or two routes with losses when all others are profitable; you cannot survive when all your routes are unprofitable. Initially, stick with lucrative yet underserved routes — it's common to find that people will sometimes ignore routes with huge demand, which you can exploit. I'd recommend only facing another competitor head-on when you are of a large enough size, so that way one or two loss-making routes because of competitions won't overly impact your bottom line.
Some would disagree with me. Some would say that direct competition is a good way to serve markets with large amounts of demand when few routes have sufficient demand to warrant flights. Others would argue that not competing at all allows your airline to slip "under the radar" and grow large, then wiping out any competition. Neither is wrong — the key part of Airline Empires is to find a strategy that works for you. And it's not the same in every market or hub on the globe, so know that each world and country work differently.
Hopefully this helps you. Good luck and enjoy playing!