Ah I've counted my pup as my personal item, since he's fitting underneath my seat. I have a wheeled carry-on that I put in the overhead bin. However, lately I find it easier just to check the bag. Lugging around a pup is enough trouble with security, etc.
Also, just some logistics FYI that surprised me the first time I flew with a pet. For American at least, you must check-in with an attendant and pay then, getting a special ticker for your pup. Often this takes forever because everyone in that line, even if it's only like 3 people, need some major help (ie flight cancelled etc). Depending on the ticket agent is how much grief I get for trying to bring my dog aboard. It's a 20 or 25 lb weight limit and depending how surly they are, they'll question my 16 lb Hav as "too big" and make me 'prove' he's the right size ('well his weight's okay but he looks a little cramped' 'it's just a lot of fur!'). I've once had to get manager to come and approve my pup. So the check-in process can be anywhere from 5 to 45 mins.
During security is another fun time, because after you wrangle your pup into the carrier for the ticket agent, you get to take him out and carry him through the metal detector (so no collar on!) while the bag goes through the x-ray. This is even more fun if you travel with a cat who freaks out with all the loud noises and new people; I've gotten bad scratches trying to keep my grip on her. For me, the hardest part is getting Marlowe back in his carrier after walking through the metal detector. It doesn't help that half the people want to say hello to the cute pup and the other half want you to hurry help and get out of the way in the security line.
As for planning, I never feed Marlowe before the flight (he gets motion sick and once is more than enough of cleaning puke out of the carrier in the airport bathroom for me). I make sure he pees in the parking lot at the airport. I always fly non-stop. I let Marlowe out as soon as I exit security and let him pee outside before I go pick up my bag or anything. You can have your pup on a leash in baggage claim
Oh the actual flight - not a big deal. Marlowe never barks anyway, so at most he'll whimper as we take off - a finger or two in the carrier helps keep him calm. After that, he's fast asleep. If you sit towards the back of the plane, any noise your pup makes will definitely be drowned out. Most people are like 'oh! i didn't realize you had a dog this whole time!" as we deplane.
Flying is a real headache. At this point, I'll drive with my pup if I'm going anywhere for any length of time (ie >1 wk). For short weekend trips, I think we're both happier if Marlowe is boarded (kennel free place) or stays with a friend.