Will you need her seat on arrival? If so, take it on the plane but make sure it is FAA approval as not all seats are FAA approved. I would not check a seat as luggage. If you don't need the seat on vacation, you might want to leave it behind for ease of travel.
How long is your flight? We probably have 50+ segments under our belt and I have learned less "stuff" is better. Getting to the airport as early as possible also helps the general stress level. I never had a problem or issue with TSA. In fact, there were times where the agents gave my son stickers.
Double and triple check your boarding passes to make sure you are seated with your child. Just because you may have been ticketed together, equipment changes happen and seats get shuffled. FAs are not required to ask passengers to switch seats (I have seen some flat out refuse to help parents) so do not walk away from the check in counter without boarding passes that show you sitting together. A 4yo on her first flight is a very valid reason for insisting they move someone else. The early you check in, the better the chance that there won't be an issue.