We've been traveling with my son since he was about 3-months-old, and there are new tricks we learn every time to make it easier. The best advice I can give you is to relax -- and bring things to entertain your child! We now travel with a portable DVD player, but when my son was younger (he's now almost 3), I made sure to pack a few (new) books, and a few (new) toys -- things he hadn't seen before that would capture his interest. I also brought a sippy cup (you can bring an empty one through security without a problem and fill it with bottled water on the other side), and snack containers (goldfish crackers, etc. are usually available for purchase once you're past security as well).
We started buying my son his own seat on the plane when he was about 15 months old -- it was just easier on us to have him strapped in his carseat than it was to try to keep him contained on our laps. But before we started doing this, we always gate checked his car seat -- it doesn't count against your checked luggage limit and you simply need to get a tag for it when you reach the gate for your flight (you can do the same thing with a stroller). I would definitely bring his seat because while you probably won't need one for taxis (it's not ideal but it's also virtually impossible to install a car seat quickly -- or occasionally at all -- in many cabs), you'll definitely want your car seat for when you rent a car. Some places will rent car seats, but in my experience, it's easier and more comforting to just bring your own, which you and your child are already familiar with.