Finding Bargain Air Fares - Web Sites I Use:
An excellent article on finding cheap air fares that says all the things that I would say: https://www.nomadicmatt.com/travel-tips/how-to-find-a-cheap-flight/
My recommendation: - only book directly with Airlines or other highly regarded Online Travel Agency. Avoid "aggregators" (see wiki article here) unless fare is clearly worth multiple hassles (no instant confirmation and challenges in (1) getting seats assignments, and (2) connecting passport, mileage and TSA security information to reservation).
From personal experience - double check airports for connecting flights (I was booked on flights "connecting" at London Heathrow only to later notice that the "connecting flight" was a London's Gatwick airport (a $30, minimum hour-long bus ride). In my excitement over finding a very cheap fare, I failed to notice the change from LHR (Heathrow) to LGW (Gatwick) with the absolute minimum (3 hours) between flights.