I picked up a new puppy from our breeder at TUL just a week ago. Our first puppy from this breeder was shipped without any incidents, but he was pretty frightened when he arrived. This time I decided to fly to Tulsa and bring the puppy back with me in the cabin. All in all, it was much easier than I expected.
I made the pet reservation over the phone after ticketing myself online. Be sure to call AA reservations before ticketing, because as stated earlier there is a limit on the number of cabin pets allowed per flight, per cabin. The per-cabin part is important if you are attempting to upgrade; I found this an excellent use of miles to upgrade at ticketing, rather than hoping for sticker upgrades and cabin-pet space before each segment. The pet fee is paid at the airport when you check in. Keep in mind that the pet record locator is attached to your own record locator. This will prevent you from using online check-in or reserving exit-row or bulkhead seats, even on segments where you don't have the pet with you. Also remember that your pet carrier counts as one of your carry-on / personal items.
Despite all the requirements, I wasn't asked to show health records or the cabin-pet receipt at any time - either to AA or the TSA.
Going through security is fairly simple; just take your pet out of the carrier and carry him or her through the detector. It can be a challenge juggling carrier, briefcase, laptop, boarding pass AND puppy - but I'm guessing you'll find like I did that you get more assistance than usual from both the TSA and your fellow passengers. Who doesn't love a cute dog?
The flights were actually the easiest part. After take-off, I pulled her carrier closer to my feet and opened up the end flap so she could see out. My puppy slept almost the whole way and when I got up to leave the aircraft most people noticed her for the first time.
Good luck with your travel!