• Know the "24-hours rule". The U.S. Department of Transportation requires "each U.S. and foreign air carrier that has a website marketed to U.S. consumers" to allow customers to cancel their reservations within 24 hours if the reservation is made at least 7 days in advance of the flight departure date. This rule applies to carriers only, not travel agencies. Therefore when booking with a travel agency (online or brick and mortar), check their policies first.
• When traveling to foreign countries check your passport validity. Different countries have different requirements regarding the passport validity beyond departure date. This is not to be taken lightly, I have seen people stranded in Europe because they were allowed to board the flight in the U.S. but denied to board the connecting flight due to their passport expiration date. They didn't even have the option to skip that travel segment due to the airline policy on cancelling their return trip.
• If you fail to board a flight segment, the next segments will be cancelled. This is one good reason to book one-way flights instead of round-trip, that way you can separate the return flight. No matter the destination or time of travel, always check both options: one-way and round-trip prices for award travel or paid fares.
• Usually, the lowest prices for airfare are found Sundays and Tuesdays. Most airlines drop their prices on these two days of the week.
• If your one-way trip involves one or more connections, look for code-share ticket (when one airline operates a flight that shares with another airline and both airlines sell tickets for that flight) or a multi-carrier, interline ticket (which allow you to travel on a single ticket, that you buy in a single transaction because airlines have an agreement between them but don't advertise tickets through each-other). Don't use separate tickets for each segment (unless you don't have a choice). With code-share and interline (multi-carrier) ticket, if one flight is delayed or canceled and you miss a connection, the airline will have to re-book you on the next available flight. Also, with code-share and interline ticket you can check your bags from origin to destination (unless you are required to clear Customs in the first port of entry). In case you need to purchase separate tickets make sure you have plenty of time between flights, as you need to account for any delay from previous flights, claim you bags and re-start the booking process. When available, you can purchase code-share flights ticket from a single airline for your entire trip. If you are booking a multi-carrier, interline ticket; use a travel agent or an online agency that you trust.
• Before making a reservation at any airline, it is recommended that you learn about flight cancellation and flight changes fees and policies. It's better to know in advance how much a ticket change or cancellation will cost you.
• Visit US Department of State and do some research about your destination. It is very easy to find helpful information about passport validity requirements, U.S. Embassies and Consulates, visa requirements and other general information.
• You might also need parental consent when traveling alone with your children to certain countries, especially if the last name of your children differs from yours.
• Visit the TSA website at  https://www.tsa.gov/travel/security-screening/liquids-rule to refresh your information on the "liquid rules", including the rules on medication and infant/children food.
• Don't forget to pack your medication, some medication, like in the U.S., is prescription only and keep it in your carry-on baggage.
• Keep your all your valuables (including your laptop) and your keys in your carry-on luggage. If you are asked to check-in your carry-on at the gate (for lack of space), remember to remove these from that carry-on and take them  with you on board.

WHEN TRAVELING WITH CHILDREN (and not only):

• When traveling with kids, you likely will need more time for boarding.
• Dress them comfortably and have change of clothes handy.
• A light stroller might be a good idea for kids under 5.
• Pack non-sweet snacks that travel well. Get travel size hand sanitizer.
• Encourage the kids to exercise before boarding the plane to consume some energy.
• Don't forget the entertainment, including but not limited to electronics, and don't forget the headphones.
• Do some research, as some electronic devices are banned on some flights. Look into "on flight" entertaintment options.
• A small, thin blanket might be handy for kids to lie down when there are no comfortable seats available in the airport.