There are many things that I've learned over the years about traveling with my own children. Whether or not it works with anyone else's kids, that's just a matter of trial and error. But I thought I'd share my wisdom (ha!) about specifically airline travel with little ones. I'll try to make this as organized as possible.
Planning the Trip:
It goes without saying that there are "better" times to fly with kids than others. A red-eye might be fine for an adult, but really not for a kid. Layovers are boring for adults, but just magnify that times a million for a kid. Try to do a direct/non-stop flight, if possible. And lastly, would YOU want to sit on someone's lap the entire time (granted, on some flights, you're almost sitting on someone's lap anyway).
Having said that, here's what I've learned. Early morning flights are nice. Why? You get to your final destination earlier, of course, and if there are any unforseen delays, flying early allows for that. By early I mean like flights starting in the 7-9am range. Kids aren't usually cranky because they're tired, and leaving early in the morning also means that you can get away with just grabbing them out of bed, putting them in the car in their jammies.
In my personal experience, we haven't done any red-eye flights (there just haven't been any offered), and usually we do early morning flights. Daniel's first flight was in the mid-afternoon when he was 6 weeks old, and he was CRANKY by the time we arrived at our final destination at around 9-10pm. Another more recent flight (this past December), we had planned on leaving very early, but due to weather issues, we had our outbound flight re-routed, and scheduled for hours later.
Next part of this...layovers. I mentioned before that non-stop/direct flights are ideal. This should be a "duh", since having to get on and off the plane multiple times, taking off and landing repeatedly, is hard, and the layovers themselves not only add to the total length of the trip, but also add another opportunity for trouble. So be cautious of the length of the layover, if you have to have one.
Our first flight with Daniel involved one layover in Salt Lake City...it was THREE hours. Trying to bounce a baby around for that length of time was painful. There is only so many bottles that a newborn will drink. As Daniel got older, having a toddler running around might have been worse. And DEFINITELY with a toddler AND very pregnant mother...layovers really were not cool.
Luckily, several flights that we took between Reno and Phoenix were non-stop, which was wonderful. On and off...quick and easy. Now comes the shameless plug for Allegiant air...we discovered that there are non-stop flights between Phoenix and Great Falls, MT. We FINALLY availed ourself of one of these, and it was fantastic. Definitely worth having limited flight times and also having to drive out to the other airport. Even if the price were worse (which, it was fairly inexpensive), we would take it just to be able to have that non-stop.
OK...the last part here a lot of parents have done for financial reasons (and so have we!). Yes, kids under 2 can be "lap babies". And in our experience, if there is an available seat, they'll let us use the carseat on the plane (and not charge). But for some flights that are bound to be very full, it's sometimes good to just go ahead and buy the seat (if you can afford it). Daniel's first trip at 6 weeks old was OK...I held him on my lap and had Mike and his brother next to me to help with grabbing bottles and stuff. He was still at the stage where he'd fall asleep in my arms. As Daniel got older (and bigger), it got more difficult to sit him comfortably on our laps, and it also got harder to keep him still. This last time we went to Montana, we went ahead and purchased Jared a seat...and it was GREAT. In general, I would feel OK about a short flight and a child under age 1...long flights, over age one...no.
Let's just say that checked baggage is your friend, carry-on is your enemy. As tempting as it might seem to be cheap and not have to pay for checked baggage...it's worth it. Also, babies and children have a lot of stuff, but sometimes less is more...you can usually buy things when you get there.
Pack in advance. PLEASE do not pack the day you are going to leave.
It is insanely rough to go through security with kids. Mostly because babies and small children have bottles of liquid that they require, they have car seats and strollers, and depending on their age, they either need to be held or kept at bay so they aren't running around where they aren't supposed to be. About the last thing that you need is your entire carry-on luggage being unpacked and repacked while you hold a screaming child with one arm and try to pack with the other. So it's just best to CHECK EVERYTHING YOU CAN. You don't need a weeks worth of diapers and wipes in your diaper bag...really. You don't need 50 changes of clothes.
What I will say about your checked luggage is to be sure and only use like half of the space in your suitcase going (or aim for that), because you'll undoubtedly come back with more crap than you took. This is probably good advice for traveling in general! Also, take extra grocery sacks or ziplocs just to keep the dirty clothes in (and they're just good to have in general).
In your carry-on....make sure you definitely have a day's worth of diapers and wipes...depending on the kid, that could be 4 diapers or as many as 12. A light change of clothes too. Now food. Bottles are exempt liquids, but it's a good idea to have them all together (big ziploc) just because security sometimes wants to do a special screen on them. It's easy to just grab the giant ziploc and hand it over. Any of your liquids (shampoo or whatever you might put in the carryon because you didn't want/couldn't check them) put in their regular quart-sized ziploc. Put these in READILY ACCESSIBLE places in the diaper bag. You really don't want to go digging with one hand.
Do NOT overpack your carry-on either. Make sure all of the things you have to take out are easy to grab and put into the bins, and can easily go back into your bag on the other end.
On the plane:
This goes with the planning and packing...but just make sure that you are constantly keeping the kids occupied. Boredom breeds trouble. A lot of this has to do with the age of the kid and personality, but it's good to be prepared.
First...bring lots of little things. One new thing even. With an infant, binkies have worked (have at least 2 or 3 because they will end up thrown or spit out and probably in a place you can't reach). Stuffed animals. Quiet (ish) rattles. Bottles. Toddlers can do toys. Again, take several so when they get bored with one, you can hand them a new one. If you have at least 3 or 4 things, you can rotate them. We've had a media-player, toy cars, coloring books, and video games, among other things.
Food! Have lots of that too. Snacks on a plane are often limited (if you don't have to pay for them, that is), and they only come around once or twice. Rice crispies treats, packages of gummy fruit snacks, bags of chips, dry cereal in ziploc bags, and crackers are the top things I can think of. Rotate food out with the toys and stuff and you can keep a kid occupied for a while.
If kids can sleep, that's even better. But best be prepared!