Google Flights is a great search engine for quickly browsing for flight availability and low fares. It's top features are its search speed and simplicity. While Google doesn't directly book flights itself, it does provide links to do so on other websites.
The Google Flights calendar has some great features that allows you to quickly get search results while searching a wider range of dates for the flexible traveler. You can adjust the length of the trip and use the +/- feature up to 3 days, to get a list of the cheapest fares for trips within the range of 4-10 days for example.
Google Flights Multi-City tab can be used to take advantage of Stopovers.
First, you should play with finding the cheapest roundtrip dates first, using the Calendar feature. This will give you a guideline of which dates will likely work for multi-city searching, instead of searching dates aimlessly.
Once you've found some possible dates, click into the itinerary to see which city the flights route through for a layover, so that you can possibly create a stopover.
Next, switch over to Multi-City and work within the dates you found to see if you can create a stopover at a cheap rate.
Google ITA Software
Googles ITA Software Matrix Airfare Search is a more expansive version of Google Flights and is recommended for the more experienced user. The ITA Matrix search is much slower than Google Flights but has many extra features than can be used to narrow down your search criteria in greater detail.
Other Booking Sites
Stopovers are extended stays in connecting cities that are over 24 hours. *A stay under 24 hours is considered a layover*. Some roundtrip flights that stop in connecting cities allow you include a stopover in your itinerary for a fee or even no cost at all. When the cost of a stopover is free/cheap, they are a great way to parlay a vacation into a trip that also hits another destination.
Use the Multi-City feature on Google Flights or another booking website to book a flight that includes a stopover in another city.
Layovers are stops in a connecting city of less than 24 hours. *A stop in a connecting city of over 24 hours is a stopover*. Depending on the circumstances, connecting cities, and time-constraints of your travel, layovers are something you either wish to minimize or maximize.
Maximizing your layover may be desired when you have a connecting flight in a city in which you would also like to spend time and when a stopover is not free/cheap. You can still see a connecting city with a long enough layover, while still being under the 24 hour threshold. Finding longer layovers can sometimes be tricky as most search engines will generally try and sort results by shorter layover times. If youre planning on leaving the airport to check out the city during your layover, be sure to allocate enough cushion time to cover the time it takes to enter/exit the airport and security, especially if it is in a foreign country where you will have to pass through customs.
Minimizing your layover is a little more obvious and is generally they way in which most flight booking search engines sort and rank flights. Most passengers prefer shorter layovers to minimize total overall travel time.
The savings from the cheap flights that Flight Baron posts arent the only benefit you can reap. Flyers must also consider the mileage one can accumulate to various frequent flyer programs.
The rules have changed over recent years for some of the major frequent flyer programs to not be as rewarding to the cheap flyer, however, there are still programs with highly beneficial mileage programs and accumulating any extra frequent flyer mileage benefit is a pro in itself.
Please check U.S. Passport & International Travel information when traveling internationally as some countries require visas with hefty fees to enter the country.
Be aware of current exchange rates as it can greatly affect the costs of your trip.
Here is a simple but great tool to quickly convert currency: Google Currency Converter
Just because you found an awesomely cheap flight doesn't necessarily make your trip a slam dunk. There are many other factors to consider when deciding whether a trip is a great deal.
Costs of accommodations, food, transportation, visas, taxes, etc. can be large expenses that make saving a few hundred dollars on an airfare irrelevant if not prepared for properly.