You shouldn't approach this with the same mindset that people in the UK do wrt academic success or the idea that state schools are probably poor.
On the academic side, it all evens out. The top US universities all accept hundreds of American students annually. Good school districts will have excellent standards. Do not worry about your children spinning their wheels.
Wrt the public and private divide - throw out everything you know from your British experience. Private does not necessarily mean better, or better equipped, or a higher standard of behaviour or expectations on the part of the parents.
Public schools in the US are organised by School District. Each district has a lot of autonomy and raises the taxes it takes to run the schools, by and large, within the District. For instance, my local high school District has an annual budget of about $80m, 97% of which is raised by local taxes, mainly property taxes. For the vast majority of schools, the resources available, small class sizes, sports and arts facilities, and standards expected will reflect the financial profile of the District - wealthy districts with high house prices will generate high levels of property taxes and will expect much from their schools, and the schools in turn will be excellent, with facilities and support services that will blow your mind. SEN provision in particular is frequently astronomical.
School Districts are generally very strict about enforcing residency requirements - you must live within the district in order to send your children to the schools in the district. They will require that you furnish proof of residency and will tell you what docs you will need. Sometimes a district will have more than one school within it, and street boundaries are observed within districts. In that case, you go to your assigned local school - you do not get to choose in-district. My local elementary school district has about ten elementary schools, each serving separate areas of the municipality. In addition, some larger, city districts could have magnet schools designed to offer superb facilities and staff in certain areas - STEM, arts, music, agriculture, engineering, etc. Some have academic magnet high schools with acceptance by very competitive exam.
Your oldest child would probably be on the cusp of entering middle school, and your younger ones would be still in grammar school (aka elementary school).
Private schools tend to be religious schools run by various churches or parishes. In my area there are Lutheran, Muslim and RC elementary and high schools. There are Jewish preschools. In addition there are Montessori elementary and middle schools, and one very small Latin rite RC elementary plus high school campus, plus one small Christian (fundamentalist) elementary/middle school campus. Private schools (at least in my area) tend not to cater for students with SEN apart from high functioning aspergers or autism or physical issues.
People send their children to a private school for different reasons from reasons they might send children to private school in Britain - mainly to do with belonging to a particular faith community or seeking a certain philosophy in the education their children receive. Private schools in general do not have 'districts'. Some parishes may not accept non-parishioners, depending on how subscribed they are. All are fee paying. All public schools are free except for the matter of property taxes, but you pay property taxes even if you choose a private school. You will pay directly as a homeowner or via rent if you rent.
Many people who move find a list of schools first, then look for a place to live within those school districts or within the in-district catchment for a multi-school district.