Good luck. Airline price control is a field of economic theory unto itself. Nobody really knows how it works except the people who do it.
If you want a good deal on airfare, you have 3 realistic options:
1) Fly a lot on expensive full-fare tickets when someone else is paying. Save up frequent flier reward miles. Spend the miles on free tickets for yourself/family/friends.
2) Be willing to take unpopular flights. These are often last-minute bookings, undesirable flight times, long layovers, alternative airports, or boring destinations where airlines park their planes for the night. Or all of the above.
3) Be a ruthless internet shopping whore. Use a website like Kayak.com (plus Southwest.com and other discount sites directly). Buy the cheapest flight you can find, don't look back. Priceline can result in some good deals, or you could be scammed into a very undesirable time vs. money tradeoff. A friend booked airfare with Priceline for a weekend wedding and ended up getting overnight layovers on both ends...which would have missed the wedding AND put him back late for work. Ended up eating that airfare and buying a "real" one. But I've also scored some deals on Priceline when I had the freedom to be flexible.
I have a friend who dated a pilot, and therefore was able to fly for free whenever there were empty seats on a flight. Good perk, she had a lot of fun weekends around the world. Might as well date someone filthy rich though, and benefit from free money/everything rather than just flights.
You can often book an unpopular flight time, and then arrive hours early at the airport and see if there are any standby seats available on earlier flights. Often, this will get you travel on the more desirable flight for the price of the cheaper flight...but there is a clear risk of wasted time, and it isn't worth it, to me. Plus, if you are asking about airfare, chances are you aren't experienced enough to be comfortable navigating/negotiating to actually get a good standby seat yet.
Pretty much all airlines in the USA are safe to fly, due to heavy federal regulation. However, I've been caught (for example, my honeymoon) when reputable airlines filed bankruptcy and actually went under, after I bought the ticket but before my travel dates. So buy with a good credit card, and this risk is more or less mitigated. Airfare might seem expensive, but USA-based airlines operate flights surprisingly close to the cost of the flights themselves. A $300 ticket to fly halfway across the country in 3 hours is a pretty awesome deal when the jet fuel costs half that...plus pilots, airports, etc. If you adjust for inflation, airfare is actually one of the few purchases that modern Americans "win" on relative to cost in previous generations (along with automobiles, strangely).
Also, consider the TOTAL cost of flying. You must arrive an hour or maybe 1.5 hours early to get bags checked (if necessary) and through security. Then there might be layovers. Each airport you experience carries a surcharge and security fees. There are potentially taxi rides or parking fees at source and destination locations. Airports like Chicago O'Hare might SEEM cheaper, on the basis of basic ticket fare...but the time and money cost of getting yourself through that airport is often higher than alternatives, especially when you factor in the risk of bad weather and the sure-fire runway delays that follow.
*Soapbox* The environmentally-conscious thing to do is not fly. Airplane travel is probably the single worst thing a person does to negatively impact the environment today. Carpool, take a train, etc. I realize sometimes time/convenience make flying the only realistic option, but wanted to put it out there. */soapbox*
