Your choices depend somewhat on the size of your child.
A child needs to be at least 40lbs. in order to safely use the seat belt provided on airplaine if it is snuggly buckled over the hips.
If your child is over 20 lbs., and within the safety limits of your seat, she can sit facing forward in the convertible seat. However, depending on the seat's weight limits, your child can sit rear-facing up until she is 30-35 lbs.. Rear facing is the safest option, so you should consider that carefully if it is an option for you.
There is an alternative to bringing a car seat if your child is too small to use the seat belt provided by the air plane, however.
The FAA has approved a device known as the CARES harness. This device is approved for children weighing between 22 and 44 pounds. I haven't tried one, or seen one used, but it certainly seems easier than lugging a car seat onto the plane (many airlines will apparently check children's car seats through baggage free of charge, although with the recent move to charge for most checked luggage, you should check with your airline prior to leaving for the airport).