Try the search engines such as Kayak and Skyscanner. Unusual combinations may bring a surprise in the fare, for example Singapore Airlines from Manila to Johannesburg return is cheaper than Singapore to Johannesburg return direct, despite being 3.5 hours longer each way. Emirates Brisbane to Dubai is dearer than Brisbane to London despite being 7 hours shorter, reason being if someone only wants to go to Dubai there's a reason for it and they'll pay, where as if they want London they can easily fly on another airline.
A direct flight is usually more expensive than a connecting one, but not always. Sales take place in various cities which could make a different routing worthwhile. An increase in capacity usually results in some good promotions, I recently got an unbelievably low fare because the airline concerned had added another daily flight on the route and needed the seats filling.
Bangkok and Cairo are cheap places for business class seats, London to New York has incredibly expensive business class but this gives the airlines their profit so they can be more generous with the economy fares.
The key is doing your homework, shopping around and being flexible.