T.H.
I agree that you'll need a carrier of some sort for the baby...a wrap or Bjorn. You'll have to actually hold them during take off and landing, but you'll need your hands free at times, especially boarding and security, so you'll need something. As far as car seats, I generally like to take the car seat aboard, but if you're alone then maybe not. You'll have to decide if you think your 2 y/o can handle sitting in the regular seat. If you think she'll do fine and won't try and get out, etc. then just check the car seat. If you think it will be a nightmare and you'd rather strap her in and forget about it, then ask a flight attendant to help you take the car seat to your seat. There will be plenty of people willing to hold a baby for a few minutes while you strap her in.
Gate check the stroller so you can unload the baby ASAP. If you have a double, I'd probably take that b/c once you are getting your luggage you'll want to strap her in so she doesn't run away!
All the baby gear/seats will be free even if you check them as baggage. Get a cart to carry all your stuff to the curb...if possible have someone pick you up there, drop off the kids and then go back in to retrieve your stuff.
The baby won't need much entertainment. I'd take maybe a few toys, some snacks, bottle, etc. but the baby will probably sleep most of the way. If your 2 y/o will watch TV take a DVD player. Play movies the whole time if you must and don't feel bad about it!!!
Anyway, hope that helps! Enjoy your trip!!
** I wanted to add that they actually loosened the rules on the baby stuff...you can take milk/formula and sometimes even water onto the plane if it's for a baby now but you have to disclose them before they go through the scanner. I've done it before recently with my son and I had no problems!