This question always seems to be a hot bed issue.  Bottom line, the best way to overcome travel delays when flying in particular is to plan ahead.  The experts say you should be checking weather conditions a week leading up to your flight, and you should have alternative routes in case you need them at the last minute. Also, make sure to have all the airline flight information readily available, including important phone numbers and apps.

One way to increase your chances getting on the next flight is to avoid checked bags.  If you must check your bags, make sure to have a carry-on with one night's worth of clothes and toiletries.  A phone charge is another essential as well. 

Airlines used to adhere to Rule 240, or the Civic Aeronautics Board regulation, requiring airlines to put passengers on the next available flight should a mechanical or service-related delay occur, even if it's on a competitor's flight.  Though this policy was done away with in 1978, many airlines still follow a form of it.  So it can't hurt to bring it up if you need to do so in a bind.

Once you're discovered your flight is delayed, it's advised to get to the service desk right away.  While waiting in line, call your airline directly or perhaps the travel agency you've worked with to book your flight.  And don't forget to act responsibly and adultlike.  Screaming and yelling won't help you get to your destination any faster.