Not all airlines will allow you to "wear" your baby in a sling or Bjorn-like device (call ahead and ask), though you can certainly use one in the airport. You will have to take the baby out when you go through security. The excuse one airline gave us is that they'd have no way to get the baby out in the event that I should become incapacitated. The Bjorn got stashed overhead that time, and I was glad that I had The H there with me to share holding duties. We put the (very) little guy into it when we were in the airport tram on the way to baggage claim.
Since then, we have used a travel system, a simple umbrella stroller, a baby jogger, or the little guy's own two feet. Yes, that's right, we didn't bring a stroller on our most recent adventure. I don't recommend that until at least age three. :) Our favorite method is the umbrella stroller, but if your child is not old enough to support his/her head yet, obviously don't go this route.
If you are traveling with a car seat, I recommend purchasing a sturdy bag for it (Babies R Us sells some) and checking it with your other baggage. Most airlines do not charge for this, as it is considered an "assistive device" for your little one. The ticket agent should have plastic bags that fit over the seat if you don't have your own, but given the scratches and stains our traveling bag has sustained, I am glad we had the seat in something other than thin plastic. If you would rather gate-check your seat, or bring it with you in the hopes that there is an empty seat on board for your little one to use *with* his/her seat, that is your prerogative. It is just one more thing to carry and keep track of.
Before you get to the security line, have all of your TSA-approved liquids-and-gels bag easily accessible, and take your jackets and belts off and put them in your carryon.
If you are traveling with another adult, send him/her through security first, with the stroller and infant seat, if this applies to your situation. Then let a few people go ahead of you so the first person has a chance to get his/her things together and get the stroller set up and wait for you at the other end. Then you will only need to slip off your shoes and baby's, put your carryon and liquids bag in a bin, and walk through with your child. You can hand the baby to the person waiting with the stroller, allowing you two free hands to collect the rest of your belongings. You cannot set the baby in his infant seat while the seat is on the conveyor belt. Guess how I know?
Once you're through security, the rest is easy. Purchase milk if your baby will need it during the flight, since most airlines don't offer it during the drink service. Fill your reusable water bottles or buy fresh water so you can stay hydrated or shake up a bottle of formula when you need to. This is much easier than having to wait for a TSA agent to test your pre-made bottles in the security line. Find your gate, and as soon as you see a gate agent there, double-check that you are seated in an appropriate row for travelers with a lap child.
If you have time to kill before your flight, let your toddler walk around as much as possible to burn some energy--and maybe work out a poop so you don't have to change one in the postage-stamp-sized airplane lavatory! If your little one isn't old enough to walk, lay out a receiving blanket for him/her to stretch out on with a few favorite toys.
When it's time to board, you can do one of three things:
1. Board early, when they call for travelers needing extra time or assistance. You can get settled in your seats without being in a crush of people, but you will be on board longer.
2. Board with the other passengers, as you would if you were traveling by yourself. If you are traveling with someone else, have him or her hold the baby while you stash things in the overhead bin and beneath your seat.
3. Send your partner on board with *all* your stuff (minus baby) when boarding begins, and let him/her get the bags settled, pull out your water bottles, and maybe even unload a few books or toys to stash in the seat pocket. Then you get to stay behind and board as the VERY last person with your little one in tow. This gives him/her the most leg-stretching, non-confined time possible, and assures that you aren't trying to juggle gate-check items, carryon, baby, and boarding pass(es), as well as trying to sit down without banging either of your heads.
Once you're on board, relax. You're not going anywhere for awhile. :)