When searching for and booking cheap flights I go to Google Flights.
Google Flights lets you search for great deals on airline tickets and you can get them pretty cheap through them.
Google Flights is a great service to use and they have a list of cheap airline tickets and when you buy the airline ticket after searching on Google Flights you are then taken to a third party website to actually buy the airline ticket.
Google Flights does not actually sell the airline ticket but they help you find great deals on airline tickets.
I use Google Flights whenever I need to book a flight and it's been wonderful.
Kayak for flights is also another good service when searching for cheap flights.
You can also usually get last minute airline tickets cheaper as well so check with your airline as I sometimes get cheaper airline tickets for flying when I book the flight last minute which I sometimes need to do.