@cat64 here's a little about schools in the U.S.
In general there are two types, public and private (and now Charter schools but I'll address those further on)
Public schools are free and open to everyone. School is mandatory after age 5, and your child must be 5 by September to be enrolled in Kindergarten. Depending upon the school and district, some Kindergartens are half or full day.
Some cities/schools offer preschool at age 4 for half a day but this isn't universal.
In general - each type of school works like the following regardless of public/private etc:
Children attend Kindergarten then 1-12th grades, K-4 is usually at an elementary school (primary school) then comes middle school which is generally 5-8th grades, then highschool, which is 9-12th or breaks down into Freshman, Sophomore, Junior, Senior.
Juniors and Seniors take the ACT which is a standardized test usually used for in-state Universities and in their Senior year, the SAT, which is a standardized test used for entrance to all Universities. Often, students in the 11,12th grades also take AP classes (Advanced Placement) which are essentially college classes that after the class you can take a test and your score determines how many college credits you have in that subject - so it's possible to start University with a number of credits or even 1 or 2 years into a degree program.
Here if you live in a catchment area of a public school, you are automatically admitted if it's your local school.
In large cities like Chicago, we also have public Magnet schools - for those there is both a testing and lottery system to see who gets admitted.
Private schools often have a wait list, and use entrance tests and require tuition fees. These can be college prep, religious based or focused around a theme like the arts. Fees for these range but are comparable to the UK.
One thing to note in the US is that even though your child can go to their local free public school - you might not want them to. For example, our local school here in Chicago is abysmal, so we'd enrolled our children in a private fee paying school.
An alternative that has become more popular in the last few years are Charter Schools. Basically a business comes in and sets up a school - and runs it like a business. These can be public (free) or private (fee paying)