First, do your research: are there budget airlines unique to the country you're flying out of and where you're headed to? Booking with a budget Australian airline (Jetstar) from Sydney to Honolulu, then an American one from Honolulu to Montreal saved us over $400 each when flying back from Australia to Canada earlier this year. This allowed us to create a thrifty five-day stopover in Hawaii on our way back, which was less exhausting and a lot cheaper! Kiwi.com and AirWander are both great search engine for revealing cheaper routes like this that involve multiple airlines.
In Google Chrome or Safari, incognito is enabled by hitting Command (or “Control” if using PC), Shift, “N”. For Mozilla Firefox or Internet Explorer, hit Command (or “Control” if using a PC), Shift, “P”. This will open a new browser window where your information is not tracked, thus not inflating prices as you search. Note: if you're using an older version of OS X, open Safari then click “Safari” in the menu bar, and select “Private Browsing”.