First of all, RELAX! Kids take to flying at an early age way better than first-time adult fliers.
DS2 is an expert flier. DD took her first flight at 8 days. They don't give wings any more for first flight...but she did get frequent flier miles :-D
There is a chat board devoted to all things flying (the founder live in Colorado Springs) and they have a forum specifically for traveling with children:
http://www.flyertalk.com/forum/forumdisplay.php?f=221
Not ALL airlines give the 50% discount for "infants" (kids < 2 in airline terms) who occupy a seat. Delta, Southwest and Frontier WILL offer the discount but you usually have to call the reservation number to get it. I do know that you can get the discount on DL and F9 by using an online service like Expedia or Orbitz. The airlines' own website and Travelocity don't offer the discount seat.
We have always bought DS a seat and will do so for DD as well. It's safer and more comfortable for them (and our laps) to be strapped into the car seat they are alredy familiar with. Plus, we then don't have to check the car seats and risk them being damaged.
If you choose to have her fly for free as an infant in arms, be sure to bring an ID for her (passport, birth certificate) in case an airline employee challenges her age. Not likely to happen at 1 but very likely to happen when at 22 months. As a minor under 18 traveling domestically, DD does not need any ID to fly. But she will need a boarding document to get through security whether she has her own seat (boarding pass with seat assignment) or is an infant in arms (boarding card).
This is a touchy subject amongst frequent fliers...seat poaching. IF your flight is not full--which you can check via the seat maps for the flight--AND you want to risk it, there's a chance that you can get her a seat but not pay for it. On the day of departure, see if the plane still have >10 or more seats open. If so, check with the ticket agent and see if they'll let her have one of the seats. They may say to check at the gate. In that case, ask for a plastic bag (or bring your own) and take the car seat to the gate with you and ask a gate agent. If there's space, they'll let her have a seat. If not, you can then just gate check the car seat using the plastic bag. Note that the only way to make sure she has her own seat is to buy her one. Plus, if you buy her a seat, she gets her own baggage allotment :-)
Kids in a car seat MUST be seated at the window and CANNOT sit in the exit row nor the 1 row fore and aft of the exit row. Window seat only is because in an emergency, they don't want the car seat blocking egress for another pax. (If traveling with parents then they usually allow the child to sit in the middle seat). The reason for the exit row is obvious but the rows fore and aft is because the person opening the emergency door will have to toss the 40+ pound door someplace and it might land on the child.
How are you planning to transport her through the airport? If you are using a convertible car seat, this gizmo [ http://www.gogobabyz.com/products/gogo_kids.html ] turns it into a piece of rolling luggage. DS has one for his Britax and it's so much easier than having to deal with a stroller.
If you do bring a stroller, stick with the smaller ones like the umbrellas or Maclarens/Chicco/Pliko Lites rather than the giant Graco Quattros. Strollers will have to be gate checked but TSA gets really cranky if an item won't fit through the x-ray machine. Mind you they CANNOT refuse to let you bring the stroller if it's too big for the x-ray, they would have to hand check.
Speaking of TSA...YOU are the parent of your child, NOT some TSA weenie. (I'm not very fond of TSA if you can't tell :-) No matter what some bozo tells you about leaving your DD while they check something, use your judgement and keep her with you at all times! Also, you are allowed to carry her through the metal detector (just remember to take her shoes off her feet). You are also allowed ANY amounts of formula, juice, breastmilk, drugs (baby Tylenol etc) in your carry-on and diaper bags. The 3oz rule will apply to YOUR items like shampoo and lotion but they DO NOT apply to BABY items like baby lotion, formula etc.
RELAX!
When you go through the checkpoint, a TSO (screening officer) may notice the baby and ask if you have any liquids like milk, juice etc for the baby. Go ahead and confirm that there are liquids in the bag and you'll probably be pulled aside for a bag check after the x-ray. ABSOLUTELY ask the TSO who is going to rifle through the bag to please put on a new, fresh pair of gloves! (They may grumble but they have to.) But if nobody asks about liquids, don't volunteer the information! Our last trip out we had 3 bags with liquids but they only asked about the diaper bag. They checked that but nary a peep about the other two bags wich had way more stuff in them! (The x-ray doesn't pick up liquids but sometimes can pick out the higher density caps on bottles.)
Your shoes. You are supposed to remove shoes when going through the metal detector. You don't have to (especially if you're afraid of slipping while carrying a baby). If you don't they'll send you to secondary screening for a pat down and x-ray the shoes anyway. Best way around this is to wear shoes you can easily slip off (wear socks!) and then slip on/step into while holding the baby. Send the shoes and stroller through first so you'll have a place to set her down and can then grab the bags.
The advice about getting new (quiet) toys or books to entertain her is very sound. Many times we don't buy a new toy for DS, we just quietly pack away a toy the week or two before departure so that when he gets it on the plane, it's "new" again.
Air pressure. Have a small bottle/sippy prepared and get her to drink from it during the first and last 10-15 minutes of the flight to help with the pressure changes. At her age she may or may not feel the affects much.
Yes, bring extra diapers just in case. Most planes will have a changing table attached to the bulkhead in the lavatory. By all that's holy do not change her diaper on the plane seat!!! Yes, some people have done that.
Some people advocate medicating kids to calm them for the flight (overseas, many flight attendants carry a bottle of cold medecation for just that!) I'm not a fan os this practice but that's up to you.
Again, RELAX!! If you're a nervous wreck, she's going to pick up on that and will be nervous, cranky and scared. Remember, you are the parent, just because you're at an airport doesn't mean you have to allow people to do things that you wouldn't allow if you were at home, in the mall etc. If you have any problems with TSA ask for the screening manager or even an airport police officer. Finally, RELAX! Just treat it like an extended trip to the mall. Have a great flight! :-)
PS if you do get her a seat and you're going to be flying much, sign her up for a frequent flier account!