Sparteyes descriptions was great-- but here is a big note– schooling is local.. each town or school district gets to set a lot of rules, with a state guideline... Like 180 days in the school year is a state minimum, but some school districts (say up Faldages way, or even out by Sparteye,) might schedule 190, or 200 days.. so that they can also have snow emergency closings... NYC usually on has 185 days in its school calendar, since we rarely have enough snow to close down the roads/schools.
and each local school district can choose its own text books -- but must meet a state standard for curriculum but there is a large leeway..
In NYC, there is Local control-- a big political issue.. but there are about 40 local school districts. they each get a share of state/city school funds basis on enrollment.. and they can figure out how they want to spend it. so some local schools have special curriculums.. they might offer journalism classes starting at first grade. or science classes with special labs, or language arts classes..
and these are open city wide.. so if your child has an interest in music, you can enroll them in a school that has a special focus on music.. the district get extra money for each out of district kid that enrolls, so each district tries to have one conveniently located school, that is a Magnet school. (drawing students to the school as a metal filings are drawn to a magnet.)
High schools are often in a separate school district. In NYC all HS are open to every student. some high-school, Stuyvestant, Bronx HS of science, Art and Music, Art and Design, and others have qualifying test to get in.. competition is fierce. as a ratio, its harder to get into Stuyvestant then Harvard.. (ratio of # applicants to # of admits)
my son went to "local HS" -- about 1.5miles from our house. my daughter commuted 15 + miles each day to Manhattan to specialty HS.
In less densely populated areas, several school districts might band together to have 1 large high-school. Jim the Dog has touched upon the problems of small HS districts.. You might only have 90 or so kids on a grade.. And total enrollment of 300 to 400– not enough students to have offer both spanish and french.
NY HS offer spanish, french, latin, german and russian..and maybe others... (or maybe different choices now.. But usually a very large assortment. But each HS usually only offers two.. If you want german, there might only be 5 schools in all of NYC that offer german. –
and about university/college levels most states offer at least some very inexpensive Jr colleges, (Community Colleges) 2 year programs.. These will sometimes feed into State colleges.. State colleges have varying fees, (cheaper for in state students than out of state) Until 20 years ago, CUNY– City University of NY was total free.. Its not now, but it is still very inexpensive. CUNY alumni include many Nobel prize winners, and pulitzer prize winners , etc. .