When I first started planning my two month journey throughout South America and Europe, I spent two entire weeks hunched over the dining room table with little sheets of paper scrawled with prices all over researching the cheapest flights for my entire trip. Here's the kicker, I didn't even book all my flights! I only booked TWO flights; my departure and return. I had to explore a handful of different websites to find the proper prices, but here's a rundown for you.
If you've got loose plans, and have a slight idea of where or when you want to go, then I suggest using Hopper or Adioso to scan for flight deals and flight trends. With these sights, you'll be able to explore flight costs from your airport to anywhere. You can even use a randomizer in Adioso. With Hopper you can look to see when are the cheapest times to go to a specific location. (I prefer to go directly before or after peak season) You'll be able to find flights through both these websites, but I like to use these prices as a starting point.
From here you can use the classic search engines like: Travelocity, Orbitz or Expedia to compare prices. I try to use a mix of Skyscanner and Momoondo as well. The truth is that search engines do not include all airlines, sometimes budget airlines like Airberlin, Spirit, Vueling and Ryanair are left off. Look up what budget airlines fly into the city you want to go to. Depending on which airline has the best price, MAKE SURE YOU VISIT THE AIRLINE'S ACTUAL SITE TO CHECK PRICES. With search engines you get charged a fee, and once in a while they end up more expensive than the actual airline prices.
I try all the search engines described above including the ITA MATRIX website. This is a site used generally by professionals like travel agents and even other websites that you use to track flights. It's a little difficult to maneuver in the beginning, but try it out a few times and you'll get the hang of it. This is a good way of checking for affordable flights if you know exactly when and where you are going. The only downfall is that you have to actually visit the airline's website or reach out to an agent to book these flights. If this sounds like too much work, just use Google Flights. It's basically the same software but a little easier on the eye.