I have been traveling as the solo adult with my dd's since they were 1 and 3 and it isn't so bad. Just remember how much you do on a day to day basis with the two of them and try and relax.
A sling or snugli for the 1yo will help especially if he/she isn't walking yet. You can't wear baby through security but it will help for just before and after when you have to have the stroller folded.
More info will help, are you bringing a stroller? Single or double? one carseat or two ? etc.
But - making a couple assumptions here is what I would do / did:
I would start out with the 1 yo's carseat in the stroller, backpack underneath and wear the baby and let the 4yo walk. In this case I would seriously consider a "hand holder or leash" just in case although I wouldn't normally use one.
Hopefully whoever is bringing you to the airport can help you check your bags curbside or park and help you in. If not definatly find a sky cap and have some ones handy to tip. I would check the 4yo's carseat through or even better send it ahead with dh to save time trying to install it when you land.
I found a single mid size backpack to be ideal because I could put smaller bags inside to separate each childs things from mine.
To clear security you are going to have to fold the stroller and put it, the carseat, your bags and everyone's shoes through (yes even the baby
).
Most aiports now have a "family line" but either way be prepared and stay calm. Have your ID and everyones boarding pass handy prior to arrivng at the checkpoint.
Once you clear security things get much easier, head to the gate and relax. You can purchase snacks and drinks after security to carry on board so you won't have to wait on the FA's if the kids get thirsty or hungry.
Ask at the gate agent to gate check the stroller, they will give you a tag to attach so you can then wheel it to the door of the aircraft fold it and leave it. It will be waiting for you soon after you deplane right there in the walkway. Be sure to fold it securely and if necessary use a bungee to keep it from opening so it doesn't get damaged.
For the 4yo drinking from a sippie cup/sports cap or chewy candy will help the ears, same for baby but maybe a bottle or pacifier and even something to eat.
HTH
TJ