If my flight is delayed and it's a nonstop:
I check to see when the next flight is to my destination and ask to be put on stand by for that flight. This is important because the next flight could leave before delayed flight depending on the reason for the delay. I find this information by checking the board at the airport. Always confirm that the flight you are trying to switch to is a nonstop as well.
If my flight is delayed and it's a connection:
I go to the airline's website and use the flight status function to check for all other flights to my destination. This way I can identify if there is a different routing that will get me there - instead of flying through Atlanta, maybe I can go through Cleveland. Be very clear with the ticket agent or gate agent that you will likely miss your connection.
If my flight is cancelled:
I do all the above plus I check to see if there is a flight going anywhere close to where my destination. For instance, if I am going to New York, I can fly into Newark, LaGuardia or JFK. I check all those options. For Boston - you can fly into Providence. For Chicago - there's two airports. If you are not sure which airports are close to your destination, ask the ticket or gate agent - can you get me close?
Also - if a flight is cancelled, a huge line usually forms to reschedule the flight. Usually there is a person directly traffic or a sign that indicate the 800 number for rebooking. While you are in the line, call the rebooking 800 number. That way you are on hold and in line at the same time. One time the line was 40 people long and I was able to rebook in 10 minutes over the phone.
If all else fails, ask the ticket/gate agent - What would you do if you were me? It's amazing how this question can open up a whole new set of options. Always be firm, but kind. Never raise your voice.
And if I booked this trip for you, definitely give me a call so I help you figure out your options.