Just FYI, there are certain seats on the plane where a carseat CANNOT go, and if you have reserved any of them, you will need to call and have your seat assignments changed.
A full-sized rear-facing carseat must be in the window position (unless the plane is a widebody, in which case you can put it in the center seat of the center section.) It also cannot be installed in the row directly in front of, OR the row directly behind an exit row. Most of the time it cannot be installed with the inboard armrest down, so that will normally have to be lifted to get it into position.
It also cannot be placed IN an exit row, but that one is kind of a no-brainer, since no one under age 16 can sit in that row.
BTW, to avoid whacking other passengers with the seat as you carry it onboard, it is best to carry it in the position of an inverted number "7" -- that is, upside-down and sideways. As a general rule, women can more easily install a carseat on an aircraft (because they have smaller hands and are usually shorter); so generally it works out best if Dad deals with the baby while Mom installs the seat.
Useful tip: due to the position and type of buckle on an aircraft seatbelt, it is important to flip the buckle-end of the belt one-half turn before you latch it to tighten down the carseat: this will insure that the lift-latch on the buckle is facing away from the carseat's shell, not toward it. If you forget to do this, it may be quite difficult to UN-buckle and remove the seat at your destination.