It's an almost universal truth in air travel: When getting from Point A to Point B, stopping at Point C will lower the price, even if Point C is 600 miles out of your way.
If you don't really mind connecting, and especially if you won't be losing half your vacation time to the layover, strongly consider it. You can save a lot of money by giving up the convenience of the non-stop.
And when you know you'll be making stops, particularly international ones, always take at least two search approaches: First, search the entire route on a couple of sites, and see what they quote you for, say L.A. to Paris with a connection in New York; then search for flights from L.A. to New York and from New York to Paris. You might find that combining two separate itineraries of your choosing costs less than booking the entire route [source: Hobica].
Up next, another point to consider in your search, especially if you live in a major city.