Most airlines' websites have a place where drastically reduced weekend fares are available. Start checking on Wednesday for the weekend ahead. When flying in the US, some of the least expensive flights are available on the smaller airlines which are subsidiaries the major airlines, for instance Delta Express and Metro-Jet. If you can't find an inexpensive flight to the city you want to go, it's often cheaper to fly to a nearby city and then catch a low cost connecting flight to where you really want to be. Depending on the distance, you might also be able to hop on a quick train or bus.
Smart travelers know how to take advantage of overbooking.
By volunteering your seat when a flight is overbooked you can earn free flights and often even cash from the airline. If you don't absolutely have to be there on time, volunteer to give up your seat, if it becomes needed, when you check in for your flight. Monday morning and Friday afternoon flights are the most likely to be overbooked. When traveling to Washington DC, always compare prices for flying into nearby Baltimore instead, it's almost always much less expensive. When flying into New York City, be sure to compare prices for flying into nearby Newark, New Jersey instead.