Well, the conventional sequence in high schools in the US is for one year (32-34 weeks, or thereabouts) of coursework as follows:
1 yr Algebra I
1 yr Geometry
1 yr Algebra II
1 yr Precalculus
1 yr Calculus
"College Algebra" is also sometimes rolled into that Precalculus course, and this is often the first time that students see some of the trigonometry that they'll need for advanced calculus topics. For the first portion of calculus, bright students shouldn't NEED anything more than Algebra II.
I have no idea how the scope and sequence compares with AoPS.
When I was in high school it was...
Algebra I
Geometry
Algebra II
Trig Semester/ Analytical Geometry Semester
Calculus
I started Algebra in 9th grade and ran out of years after Trig/Analytical Geometry and then only needed College Algebra and Statistics in college to get my degree in Psychology.
My understanding is "Precalculus" and "Trig Semester/ Analytical Geometry" are roughly the same course just with a different title. The H.S. I went to when I was at that level called it Trig/Pre-Calc.
My son is just completing Algebra II and the last two chapters are trig. Next year Pre-Calculus will include more trig, and other topics to prepare for Calculus. And some beginning Calculus at the end of the end of the year. He will be taking Calculus in 11th grade and he will be far from alone.
As to how it compares to AoPS. There books do not line up with the standard sequences. According to them the Algebra I books is what is covered in Algebra I & most of Algebra II (except the trig) courses.
As for if the AP class is a 2nd class in the subject, at my son's school it totally depends on the subject matter. Math there are prerequisites, one can't take AP Calculus without Pre-Calculus. AP History classes are less straightforward, one doesn't have to take for example US History at the High School level but it's assumed they have taken US History in 8th grade. The AP Language Course is really broken into two years course, Honors American Literature and AP Lit (that covers the English Literature). Thus 10th grade H. English is a requirement for the course. Science is a mixed, AP Chemistry and AP Biology both require a previous high school science course. But other science courses such as AP Physics and AP Environmental Science don't. And there are lots of courses like AP Computer Science and AP Music Theory that have requirements but it can be the first time you have seen the subject.
