I am not in the teaching industry, but two of my sister in law's are. One thing to consider economically, when deciding between public and private schools is that public schools will receive more funding from the state and are often more lucrative in terms of salary.
That being said, there are certainly pro's and con's to both. I would recommend shadowing and reaching out to family or friend connections at both public and private schools. The best method for me has always been to get first hand experience with someone in a profession.