Some require a roundtrip ticket, though there are some that give lower rates for one-way tickets. Most do not permit open-jaw travel (most require circle-trip for excursion fares). Some permit stopovers, and may or may not charge you for the privilege (typically $15-30 per stopover). Fares are often seasonal. 

Children's rates are usually discounted against the applicable fare. (Some airlines now apply children's discounts against the highest fare only.) As usual, children must carry proof of age. 

Note that fares are almost always not applicable to/from intermediate points. This means a ticket from Boston to Chicago passing through Pittsburgh could be cheaper than a ticket from Boston to Pittsburgh! But, of course, you can get off at Pittsburgh so long as you don't have checked bags nor have subsequent legs on the same ticket. 

Use the same carrier and flight class for all segments of your itinerary. Changing airlines usually adds to the cost of your trip. But sometimes you may be able to get a dirt cheap fare on one airline to an airport 150 miles or so from your destination, and then use another airline to get to your final destination. (This most often happens when the first airline has no direct flights to your final destination.) 

Sometimes fares which involve a connection are cheaper than direct flights. So if all the fares are non-stop, ask if flights that involve a connection are cheaper. For example, flights from Pittsburgh to Boston on TWA are often cheaper than flights on USAir, because USAir offers non-stop service while TWA flights are routed through their JFK hub. 

Other tips:

• If all the fares are on one airline, ask your travel agent if there are cheaper fares on other airlines. Be prepared to ask for specific airlines. Don't run down a list of a dozen airlines, but ask for two or three. If all show similar lowest fares, you aren't likely to do better on another airline. (Obviously, this advice doesn't apply if you're calling the airlines directly. If so, call 2-3 airlines before purchasing tickets.) 
• If there are two airports near where you live (e.g., Washington DC, New York), ask if fares from the other airport are cheaper. It may pay to drive 40 miles to save $100 on airfare. 
• If you qualify for special discounts (youth, student, senior citizen, etc.) ask about the availability of such discounts. If you don't ask, they won't volunteer the information -- how are they to know whether you qualify? 
• Tickets are generally cheapest for travel in late August and from March (excluding Spring Break) through mid-June, when air traffic is the lowest. Of course, this rule of thumb depends a lot on the destination, since some destinations have strong traffic year-round.