I'm a former Flight Attendant and yes, you can bring car seats on the airplane. They are never provided by the airline. Okay, I've only heard of one, British Airways and only if you're booked in certain seats, on certain flights, etc.
The good news is that if your children have a seat and you bring a car seat that is FAA approved, it's your *right* to use it onboard. It doesn't have to be an infant bucket seat either. This is actually the only safe way to fly with a child. Babies on laps are not protected if anything goes wrong and children under age 4 don't fit the adult seatbelts on airplanes very well. Not only are they safer, but more comfortable in their familar seat.
Most car seats in the U.S. are FAA approved. Check out this site and look on the side of your seat;
http://www.faa.gov/passengers/fly_children/crs/
Please note that I'm talking only about U.S. carriers. If you are flying an airline of another nationality, check their site and/or their country's authority (ex. the CAA in the U.K.) to find out what their policies are.
Children over age 2 have to have their own seat (true on most airlines) but a car seat is never required. I do really suggest it, not just for their safety but also it guarentees that your car seat will make it to your destination undamaged. Checking car seats is risky and it's never a good idea to rent a car seat from a rental company. They are often old, dirty and/or don't fit the child correctly. I had a bad experience once and I'm not alone!
If you go to one of the travel with children websites, they'll sell you all sorts of gizmos to get your carseat through the airport. Some attach to your carseat, others simply cover it. Please be careful that the product actually will fit your specific seat. For example, the Gogokidz only fits certain brands.
http://www.gogobabyz.com/products/gogo_kids.html
A good tip I got off the net and used with success is to simply attach the carseat to a metal luggage cart (a relic of my first airline) and attach it with an extra bungee. It worked great and I've done this while flying alone with three kids.
Since I fly a lot with my own children now, between Europe and California, plus shorter hops inbetween, and worked in the industry for 10 years before having my own kids, I get a lot of questions on the subject. I finally put together a totally non-commercial article for a local newsletter, that I later expanded and posted on a blog to share with other parents. I go more into detail on packing tips, etc.
Please feel free to visit;
http://flyingwithchildren.blogspot.com
Have a great trip!