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.
You're not crazy for thinking that a flight price has changed after searching it a few times in your web browser. Based on the cookies in your browser, flight prices do increase when a particular route is repeatedly searched, as the site wants to scare you into booking the flight quickly before prices get even higher. Always search for flights in incognito or private browsing mode to see the lowest prices.