Warning: Having already checked your baggage can make changing flights more complicated.
If an airline can't get you to the destination you booked, it will occasionally buy you a ticket on another carrier. It's rare, because the airline has to actually pay its competitor for the ticket.
It still might be worth asking the airline, especially if your delay was within the airline's control and there is no reasonable way to get you to your destination on their planes.
You can also consider going to a different airport.
Major cities often have more than one, and it can even be worth the trouble to rent a car and drive a few hours to get to a city that has flights that will get you where you want to go.
I've driven the two hours to Richmond, Va., twice when there was no way to get a flight out of any Washington airport.
Ask for compensation
After your delays or cancellations get resolved, send an email with all of your details, including your frequent flier number, to the airline, explaining what went wrong and asking for compensation. Or ask the agent who helps get you on another flight about vouchers or additional miles.
Many airlines award frequent flier miles or vouchers for future travel based on your status with the airline and how long you were delayed getting to your destination.
On some planes you can even access the airline's website for free on the in-flight Wi-Fi and submit a comment.
But note that airlines generally don't give compensation for events beyond their control, like weather delays and cancellations.