Here are a few tips in a nutshell to help you save on buying airplane flight tickets.
Use a tool - Google (Recommended), Kayak, SkyScanner and Skiplagged will compare prices for you
Save with GiftCards - Go to CardBear and get airline gift cards to save another 5% off
Sometime people are offering vouchers and upgrades on Craigslist, but buyers beware for Craigslist scammers
Buy the Upgrades on eBay - You can buy business / first class upgrades on eBay for fraction of the price. It's still expensive, but it's really at a fraction of the original price which is exorbitant
Wait for bad news - Usually after a bad PR like a disaster happening at your destination or with the airline is when the ticket is the cheapest. You see a bad news about American Airlines treating their customers? It's the time they need to lower their prices to get their planes full.
Tuesday to Thursday - are known to be the time that tickets are cheapest. If you cannot nail these times, flights are known to be generally cheaper in the hours regular people are sleeping
Use Incognito Mode - Use incognito or clear your cookies. Some flight sites purposely raise their price each time you visit them.
Choose the first flight out - you may need to wake up early, but this should give you a lesser chance of the flight being delayed from previous flights
Book as an individual - If you book as a group, the flight will try to charge you the higher price to everyone in the group
Set an alarm to checkin - for 24 hours before the flight. Some airlines will start offering special prices or priority seating for people that check-in sooner.
Find the Best Seat - Some flights allow you to pick a seat, see seatguru or seatexpert to see which seats are the best on your plane