The cheapest and fastest is to apply for a passport (I would suggest to get passport book and passport card to have as backup). It is not clear that Department of State and USCIS share information, at least not at the record level. My A-file didn't have any reference to my US passport. I am not sure about children, but assume there is no connection. Getting a certificate of citizenship (which costs $460 and takes about 6 months to obtain) creates a record on the A-file and produces a non-expiring proof of citizenship. If you are going to do this it is better to do it earlier when you have all the information you need at your fingertips, don't let your child grow and have to do this when he's forty or fifty and might become more difficult to prove that citizenship was derived. Some people might go through live happily without ever needing the certificate, however it seems that some government agencies might insist on seeing a certificate for any foreign born person, instead of just a passport. One can say that they are ignorant and don't know about derivative citizenship, but there is no denying it is a hassle to proof them wrong.
Summing up. Get the passports first, and you can also apply for N-600 certificate of citizenship concurrently (you don't need to send originals with the N-600 application) or after you get your passports. If money is an issue you can just not get the certificate and save that money for now, but your child might encounter a situation in the future in which having the certificate can save some hassle.