If you check your luggage on the initial check in, than they (American airlines) will connect them to your next flight, as long as the connecting flight is part of the original ticket purchase. If you don't have to check them, than I would just carry them on and store in the overhead. You would be amazed the size of carry on luggage you can get away with as long as you follow TSA rules and put overnight stuff in a ziploc for security check points. As for us, we never check anything, we just carry our packs on board and stuff them in the overheads
As for your connection, since you will be landing and departing into a secure zone, you shouldn't have to go through security again, although, some airports in the states have check points at the entrance to every concourse, just depends. If you are connecting with the same airline, than your gate should be in the same concourse and you should not have to do anything but deboard and go to the gate for the next flight in the same concourse to board.