At 3 weeks old, you are lucky. It's really not that bad. My son flew several times before 6 months old and these are actually the best. I have a few suggestions for you. #1 MOST IMPORTANT...let the baby suck on something...pacifier, bottle, breast during both take off and landing. The reason most babies cry on a flight is that they can't and don't understand equalizing the ears. I have seen moms refuse to give a pacificer because they are against them...this does not help anyone- not you, not the people around you and especially not your precious little one whos ears will hurt if you can't get a sucking action of some sort going. So choose your method, but have a backup...
#2 Check the carseat and go with a baby Bjorn or sling of some sort, especially if you didn't get a second seat- which at that age is a total waste. You don't want to have to lug that car seat carrier around and you will have to take the baby out and put car seat and stroller thru the xray machine. THIS IS NOT EASY. A baby bjorn type thing leaves you hands free to carry a diaper bag and pull a suitcase if needed. Plus when you get situated, you can take the baby out and easily put the holder under the seat in front of you or in your diaper bag.
#3 BORROW, BORROW, BORROW...depending on who you are going to visit...it is totally worth purchasing a pack & play with Bassinet & changing table to keep at grandma and grandpa's house. The expense far outweighs the hassle of lugging one with you, especially if this is your first child. We keep ours at my parents house and just shuffle it around to whomevers house we are staying while we are in visiting. Then, borrow things. A stroller, toys, playmat- a blanket will always do.
#4 Have whomever you are staying with pick up a pack of diapers & wipes for you, so you only need to pack enough to get you there. This is the most rediculous thing to pack in a bag...they are the same price everywhere you go...just get them on the other side. #5 If you think you may visit often, leave duplicate things behind. At both grandparents houses, our son now has a drawer. I know that there are wipes, bowls, spoons, babywash, washcloths, bibs, a towel, an extra pack and play sheets, etc... at each house. These are all the things that as your baby grows you will become obsessed with packing. So, do yourself a favor and ask for a drawer. Now both moms are conditioned to check the drawer before we come and they even stock it for us before we get there with diapers, food, snacks, new books & toys. My son loves his drawer at Nana's house...
LASTLY..if you will be traveling alot as the baby get bigger, so your best to schedule a flight that departs close to nap time. You'll be glad you did!
OK, so alot of info. Good luck to you and don't stress about it!
B.