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I haven't traveled with a 10 week old, but I have traveled with a 4 month old. Take a copy of the birth certificate just in case. We needed it with one airline, but not the other. We were able to borrow a carseat and stroller at our destination, which was awesome! They were from friends, so we weren't worried about whether or not they had been in a wreck. If you can do that, I would recommend it. My parents also bought a carseat, stroller, PnP, and infant tub for when we visit them, which is really nice. I wore LO through the airport. Luckily, I didn't have to take her out of the carrier for security or takeoff, which isn't always the case. And we nursed. A lot. LO wouldn't take the bottle... maybe because she was so distracted and in a new place?

If you want to take the carseat and stroller, definitely do so. They can't charge you for checking them. Check them at the gate though. If there is room on the plane, they usually let you use the carseat during the flight. And it's nice to have the stroller to get you and all of your stuff through the airport.

Sometimes you can rent a car seat along with your car rental. I would bring a birth certificate and at that age I wouldn't use a stroller at all. Just a baby carrier. Ergo or baby bjorn or whatever.