I hope you all don't mind me putting on my
Travel Agent hat on to provide some general advice! I work in corporate (vs. leisure) travel and I've got over two decades of experience in dealing with the airlines.
Disclaimer - The following are general airline policies, YMMV (literally) with your specific situation and airline.
Airlines typically are willing to work with you when there is a moderate to major schedule change. As many of you already noted, this includes allowing you to change your destination and/or origin airport(s) to get a more acceptable option. All of this requires that the flight(s) you want to switch to are available, if they are full or nearly full, you may not have the option to change. Having a basic economy ticket may also restrict the options you might have to switch. In any case, the bigger the change you want to make, the less likely you'll be able to do it online vs. having to call, and having multiple people on multiple reservations can also complicate matters. This isn't the airline necessarily trying to be difficult, it has a lot to do with the limitations of the system the airlines use to manage reservations and ticketing - sometimes a human just has to physically make the change.
If you need to call, trying early in the morning can mean a shorter wait, I always try calling first thing for my corporate clients. Also - if it's not super urgent - check the national weather forecast before calling. If there is a major ice storm in Texas the day you need to call, your hold time is going to be measured in hours and not minutes. It may pay to wait a day to deal with it.
One more option is that if the schedule change is big enough, you can request (and likely receive) a full refund - you are then free to turn around and book flights on a different carrier (check prices and availability on your new airline
before you cancel your existing reservation though) or postpone your trip to a later time. What's considered "large enough"? That varies by airline but typically it has to be a change of departure or arrival time of at least 90 minutes (some airlines require 120 minutes or more) OR a non-stop flight has changed to a connecting flight. Also a flight going from mainline service to regional service (even if it's still a non-stop or the times didn't change much)
may allow for a refund as well (example an American Airlines flight changing to American Eagle operated by Republic Airways). If your change is less severe but you still want a refund, you can ask - but it's not nearly as likely. You may not see an option for a refund online even with a big schedule change and may need to call to request one.
Even if you are willing to accept the schedule change as-is, you should still double check your seat assignments (for those airlines that provide advance seat selection). Often a change of schedule also means a change of aircraft type and that may require the airline to move everyone around a bit. Even if the only change to your flight is a flight number change or a time change a few minutes...double check your seats! While airline software does a passible job of keeping parties together (all of this is automated folks), it's not fool-proof - if the software needs to split parties up to make everyone "fit", it will. You'll have much better luck getting your group back together again if you do it a month out vs. at the gate because you never noticed they split you up at some point.
Don't rely on the airlines to notify you in a timely manner. It could take 1-2 weeks or sometimes even longer for them to email you the new info after they've changed their schedules. By then any good alternate flights could be gone, or it's harder to get your seat assignments straightened out. Don't be obsessive about it, but checking your reservation on the airline website every 1-2 weeks or so isn't a bad idea - and again even if you don't notice a time change, look at your seats to make sure they weren't changed. It's not a hard and fast rule but airlines typically change their schedules over the weekend, so checking on a Monday is a good way to go.