Momondo is a great site to find and compare cheap flights. Momondo always seems to find airlines that offer the best deals and their calendar view lets you see which days are cheapest to fly. I like them because they search the small booking sites no one else does. When you're searching for flights, make sure to give this site a try as well! You'll be happily surprised
No one does it like Google. Google Flights is the first place I go when I am looking for a flight. Google Flights lets you enter your departure airport and see flights all over the world in a map so you can see where the cheapest destination in. You don't book directly via Google Flights. After you select your desired flights, you will get re-directed to the airline/site to book it.
I am a huge advocate of doing research when trying to find the best deals on air tickets. Sky Scanner is another great resource. This site searches a lot of different airlines, including many of the budget carriers that larger sites miss.