Here are The Traveling Professor's airline tips you probably never heard of before:

Buy the Middle Seat:  Tired of traveling to Europe or South America crammed into a seat like a sardine, especially in a 3-across configuration?  Do what I did on my recent trip to Europe.  I bought the middle seat.  My travel partner took the window seat and I took the aisle.  Plenty of room and much cheaper than buying a Premium Economy or Business Class ticket.   Why does this work?  Let's do the math.  Two round trip economy seats between New York and Madrid on American Airlines are $485 each.  That's $970 for 2 seats.  Each Premium Economy seat is $987 on the same flight.  That's $1974.  Now purchase three economy seats for a total of $1455.  But wait - it is usually less to purchase an unoccupied seat since less taxes are applied on an unoccupied seat.  And even better, on my last trip, I redeemed my middle seat with award miles.

Buy Airline Tickets with a Travel Credit Card:  Preferably one that provides travel insurance, preferred boarding, and no checked luggage fees.  Some even have benefits like reduced mileage awards and lounge access.

Buy Directly from the Airline:  Avoid using websites like Orbitz, Expedia or any other third-party booking sites to buy airline tickets. When there is a problem (cancelled flights, overbooking, time changes, delays) those purchasing from third-party sites are taken care of last - if at all. When there are problems with a flight, the airline will often tell flyers to go back to the site where they bought the ticket from - and good luck when trying to resolve a problem through one of those on-line booking sites.

Go Indirect:  Whoever can figure out the logic an airline uses when figuring out ticketing prices should be nominated for some type of prize.  In any case, look at connecting through a city to get to a destination to find significantly lower fares.  For instance, I have a choice of flying out of New York, Boston, Hartford or Albany.  I priced a direct R/T flight between JFK and Paris CDG at over $1,000.  However, if I initiated my flight in Boston, which connected with the same $1,000 flight at JFK, it was hundreds of dollars less.