Tips for traveling with an infant

Hello all! I'm going to be flying with my baby girl who will be 6 months old at the time. I'm curious about how it works with the car seat. Do you check it as luggage? Also, I'll be renting a car where we're going, so should I bring the seat base as well, or can I just buckle the car seat with no base? Do car rental companies offer a base? Any help greatly appreciated!