Johnny says if you're planning to travel over Thanksgiving, the key is flexibility. Try a day or two before and after Thanksgiving, or if you have to fly on Thanksgiving day, fly in the afternoon. You'll get better deals than in the morning. When you search, don't put in the time of day. You'll get far more choices and better prices that way. Fly from alternate airports, which could be cheaper to fly into. Check out Google Flights first. Then go to the airline directly. Johnny Jet also has his own search tool. Southwest Airlines provides their own flight information. Stay away from low fare carriers though, because they charge for everything. Spirit is notorious for that.
Follow the airlines on Twitter. They always do special deals via Social Media. Set a fare alert with TripIt or Kayak and Watchdog. You just put in the price you want to pay and they'll send you an email when the price drops.