I have traveled alone with my daughter (now 21 mo) since she was 3 months old and it is not as scary as you may think. Everyone will be extra nice because you are traveling alone (I find security tends to be nicer if you have your act together). But be prepared that they will cry and distrub everyone else. If you are OK with that, than you'll be fine.
Double stroller would be key. You can take it to the gate and they store it for you.
NO car seats on the plane. If you cannot rent one or borrow one from a friend/family check it (or they will check it plane side with your stroller).
Carry a diaper bag/back pack with the essentials only.
Make sure your bag has wheels!
DVD play is KEY! With new videos .. and favorites.
I have found that security will let you check resonable amounts of milk. I still take thru security up to 4 bottles/sippys of milk (my daughter needs Lactose free which I cannot get in an airport)
No TIE shoes. Slip on only and if you have them in a stroller, do not get them ready at security .. but rather gather up your stuff and wheel them to the gate and then reassemble them.
If possible, pack all jackets and do not wear them. One extra step you don't need at security and bulky on the plane.
Travel the kids in PJs if necessary.
Pack something for yourself to eat!!
I usually wait until everyone has left the plane to deboard. No sense fighting everyone else and my kids when I can walk off 5 mins later free and clear and get my daugher in her stroller and get situated to start my treck to pick-up.
Good luck!