Here are The Traveling Professor's BASIC tips for getting good international airfares:

Don't Book One-Way International Tickets:  On most legacy carriers (Delta, American, United) a round-trip flight needs to be booked to find a good fare.  For example, I saw a round-trip ticket from New York to Paris for $712 but to book a one-way from New York to Paris, it was $2,727.  The exception to this rule is on some carriers like Icelandair or Norwegian Shuttle.  

Open Jaw Tickets:  Most people fly round trips from the same cities.  For example, Detroit to Rome then Rome to Detroit.   However, if starting a trip and Rome and ending in Venice, you can book a R/T ticket from Detroit to Rome and then Venice to Detroit.

Book Directly with the Airline:  Once a booking is done with a third party like Expedia or Orbitz, it just adds another layer of communication in resolving any issues with a ticket.  Book directly with the airline for best price and easiest resolution to problems.

Try Booking with a Connection:  In some (not all) cases a direct flight can be more expensive than a direct flight. So, for instance, if you want to fly to Europe from the Washington, DC area you have a choice of 3 airports; DCA, BWI or IAD.  Flights from IAD may go directly to Europe, but flights from BWI or DCA will probably require a connection.  Those flights with a connection are often cheaper.

Remember, Air Fares Don't Make Sense: My classic example is my classic example when booking a flight to Paris.   I live equidistant between Boston and New York.   I priced a R/T direct flight between JFK and Paris at about $1,100.  But from Boston it was only $750.  The strange thing is that the flight connected with the $1,100 flight from JFK! 

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