There's more than you ever wanted to know about flying with children on this site. Much of it isn't newborn specific but it might be handy in the future.

I would strongly suggest reserving a seat for your baby to fly in. It is what the FAA recommends though others have posted here that it doesn't work as well for them. I like it because once I strapped my baby in, he went to sleep and slept for most of the flight (our baby is a very bad napper when we hold him. This might not be an issue for you). I didn't keep him in it all the time but it was really nice to have the option of a safe place to put him. Particularly since coming back we had 3 aborted landings which were really bumpy. He slept through the whole thing without a whimper.

Personally, I'd also go for the direct flight. Traveling with a baby isn't as bad as I thought it would be but I certainly wouldn't want the journey to last any longer than it absolutely had to.

I've flown twice now with our 6 month old and each time was amazed at the outcome. On the last flight, my husband booked and by accident chose the flight with a 2 1/2 hour layover. I was pretty nervous about it, but in the end, it was really fine. Airports generally have comfortable seating, so we set up camp in a location that was comfortable enough to play and feed and change.

It is worth noting that (in my observation) one of the most fun things for a baby to do is watch people, especially in movement, so they will likely love the airport with all of its lovely distractions.

It might not be worth the saving when you consider the layover, and re-boarding, too.

Keeping the baby regularly breastfeeding for most of the trip always worked for ours - they slept very well and helped with pressure changes.

It gets much harder as they get older, of course.

My wife and I flew to Taiwan (quite a long flight) with our daughter when she was 5 months old. We reserved bulk-head seats and the airline provided a bassinet that hooked into the bulkhead. I think larger planes usually have that option available. A good part of the time, our daughter was strapped in the bassinet and slept soundly. Regular breastfeeding was also helpful when she was awake.

When I traveled with my kids when they were little was difficult but here's my tips. Be sure to have a pacifier, sippy cup or anything to suck on for take-off and landing. Helps with the ear popping. Buy the extra seat, makes it worth it in the end. I prefer to have as direct flight as possible because of the car seats, stroller, bags, etc. Since yours is still so young you could do either way but really, buy the seat. As kids start walking it gets more challenging since they want to walk around. If you want more little tips about traveling, let me know. Try to remember to bring hand wipes (or liquid sanitizer) to try and help ward off the cold or flu.

When we flew with our 4 month old we reserved a bulk head row with a bassinet and it was perfect - made the 8 hour flight a breeze for us!!

The answers to this similar question may also be helpful.

http://moms4mom.com/questions/208/tips-for-air-travel-with-an-infant