As someone who has worked for a major airline in pricing, the whole "tickets are cheaper on a certain day" thing just isn't true. You have two solid options:
1. Sign up for a credit card associated with an airline - most give a huge bonus award miles on signing and you earn miles every time you use it, so you will be able to get some free flights pretty quickly.
2. Use a discount carrier like Spirit - however, read EVERYTHING they send you because they will attempt to charge you in ridiculous ways for pretty much everything. Also, if one of their flights cancel, they won't rebook you on any other airlines so you will have to wait for another Spirit flight, which could be several days away.