Airlines are massive fuel consumers so if you can travel by train you should do so instead.
tl;dr: There is no Golden Rule to Cheap Tickets from here. Use all the resources below.
These are general guidelines! I encourage you to try other tactics/techniques and experiment with dates, city pairs, search engines, points of sale, and so forth.
##Airfare General Guidelines
###Frequently Asked Questions
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Can I buy a flight (A-B-C) and only fly the B-C leg? NO. This is not allowed. If you skip any leg in your itinerary, all subsequent legs are cancelled.
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Want to arrange a longer stopover or return to or from a different city compared to the one you departed from or arrived in? Look at booking a multi-city ticket. Most airline and airfare search engine websites have this. You can first search for the normal flight, determine which layover cities you want to make a stopover and try booking a multi-city with an explicit stop at that city, making it a stopover.
###General Searching
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Use ITA Matrix, Google Flights, and Kayak for aggregation. Others: Adioso, Routehappy, Momondo, Skiplagged, Hopper, Dohop.
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See WikiVoyage for discount airline info. Examples: RyanAir, AirAsia, EasyJet, JetStar, Tiger Airways, Southwest, Frontier, Peach, Jeju Air, Vanilla Air, Scoot
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Don't know when and where to go? Use Kayak Explore or the "Everywhere" destination in Google Flights Explore to get an idea. Try the Hopper Explorer.
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For students: try STA Travel or Student Universe
###Booking
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Book the flights on the official airline site (strongly recommended) or on a 3rd party booking site (as a backup) (e.g. expedia/orbitz) - whichever one is able to replicate the flights you found in Step #1. If not, call your local travel agent. We highly and strongly recommend you book directly with the airline if you can and only use a third party OTA if you can't manage to replicate the desired routing, itinerary, or price.
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Booking separate tickets can be cheaper than booking it all on one ticket - especially if you can use discount airlines. Caution: if your initial flights are delayed, your next flights are not protected - it will be considered missed - so do not book tight connections on separate tickets. See this thread for a lesson in booking separate tickets. See: General guidelines for connections. One exception is that AA will generally protect separately booked OneWorld tickets. Here's a great post on the cautions and consequences of separate ticket connections.
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Other tips on booking complex itineraries. Consider departing from a major hub (and driving there) instead of your regional airport.
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Open-ended searching: try ITA matrix if you want to search using multiple origins or destinations. Use Google Flights Explore or Kayak Explore or Skyscanner if you have a fixed origin but are interested in exploring multiple destinations. You can enter multiple airport codes in origin or destination fields in Google Flights and ITA matrix.
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Need to book around country-based travel restrictions or country-based pricing? Try Expedia Japan for Yen pricing or Expedia Canada for CAD pricing. Most airlines also offer country-specific sites. See here for more details. This is called Point of Sale.
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Be careful booking from 3rd party travel agencies. Larger ones including expedia, orbitz, priceline, travelocity all provide limited customer service and would present some difficulty if you ever need to change anything. Smaller 'discount' ones you find on Kayak and Skyscanner, including cheapoair, flighthub, etc also have similar problems. I want to avoid writing anything libelous so this is just a warning based on
###Timing, Price Information, When to Buy
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Start searching more than 4+ months in advance and monitor your flight prices every day. If the price rises, you have to commit to a maximum price you're willing to pay. This isn't a science. An example of an experiment I conducted. Explanation of "when to buy". And this stackexchange post reinforces my comments.
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Sample Data: Take a look at this thread which shows the progression of ticket prices from 2 weeks to 4 months prior to departure date. It's a single sample, but it's a pretty good demonstration of the 4-13 week rule. This study done by ARC shows that prices starts dropping at the 3-month mark, with the cheapest tickets ranging from 3 weeks to 13 weeks in advance.
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Book when you are comfortable with the price! We cannot speculate airfare prices!
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Holiday airfare I recommend you start looking around 5-6 months prior to departure and buying approximately 10-16 weeks prior. Example: the best prices for flights over US Thanksgiving near the end of November is before Labour Day (first week of September).
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Price Alerts - for example, on Kayak Price Alerts or Airfare WatchDog
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Tracking Prices - using Google Flights, you can save your favourite itinerary and Google will produce a graph tracking the price over time. More information here
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Why prices differ for the same Economy Class ticket: the general idea behind fare basis codes - the reason why round-trip flights can sometimes be cheaper than one-ways: one-ways are usually on a more expensive fare basis. /u/sataimir explains Fare Classes very well.
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Flying last minute is vastly more expensive than planning ahead. Airlines typically close down the "discount economy" ticket sale window around 1-2 days before departure. Always plan ahead if you know you need/want to fly somewhere. More here.
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"Flexible" tickets: there are tickets which are flexible, where they allow the passenger to change the dates with fewer or no fees or even change the city of departure/arrival (within same region). However, these are often vastly more expensive and for Economy class usually called "full fare economy". Different airlines have different names to them: "Flexible", "Full". Air Canada calls them "Latitude" fare. Other airlines simply use the booking class / code "Y" or "B" to denote full fare economy. They can be 3-5x more expensive than the most restrictive/cheapest economy class ticket. So you should weigh your options: cheapest round-trip + change fees + fare difference vs. two one-ways when you need them vs. full fare/flexible economy