Zoe was only three and a half week old when we flew with her the first time. It was impossible for me to find any information online about anyone who had flown with such a young baby and most people thought we were crazy. Perhaps we were, but Mark's summer contract in Sweden could not be postponed and I was happy to spend my maternity leave close to family. There are lots of well-meaning advice about not taking a newborn outside the house before he/she is at least 6 weeks, but if your baby is healthy and you have become comfortable with breastfeeding in public by then, there is nothing wrong with traveling with a newborn. As long as you follow a few key rules.
Have a small travel bassinet for the baby. A baby under 4 months needs to lie flat most of the time. The baby's spine is too fragile to sit up in for example a car seat for more than 30 minutes per day. Even though most airlines offer bassinets on the actual plane, the hours-long wait and layovers mean that you want somewhere to put the baby which is not in your arms.
Carry the baby in a sling or another carrier on your body, do not bring a stroller or pram, check it in if you need it at your destination. Up until November 2010 TSA in the US let you go through security with your baby in a carrier; after that date it is up to their discretion and 80% of the times I have flown they have asked me to take Zoe out. Even if you are asked to take out the baby, as long as you have someone to help (the other parent), it is doable. If not, this is where the bassinet come in handy: let the bassinet clear the x-ray first, take the baby through and then lay the baby in the bassinet while you gather the belongings. As Zoe got older and could sit up, I simply put her in one of the plastic tubs for bags. People usually find it funny and she enjoys it for the time it takes me to put on the carrier again.
Have extra milk (breastmilk or formula or whatever you prefer to feed the baby) in a bottle. As much as I don't mind breast feeding in public, on an airplane seat or pretty much anywhere, it is useful for two reasons: The other parent can feed a crying baby while you get something to eat or take a break, or if you are by yourself, you can give a crying baby *more* milk when your poor breasts have run out or are too slow for a baby who has had enough of sitting on a plane. Although I exclusively breastfed Zoe up to 6 months, this has been helpful for me many times.
Bring at least two sets of changing clothes for the baby and one for yourself in your carry-on. I cannot tell you how many times I have had to change Zoe's clothes due to an exploded diaper or simply her being a messy eater but the worst time was the time I forgot to bring a shirt for myself and I didn't notice a leaky diaper before... well, let me just say I called my mom as soon as we landed and asked if she could bring an extra shirt when picking us up.