American schools are divided into sections in three ways (in different school districts):
(common, and increasing)
Elementary school: Kindergarden, first grade through sixth grade
Middle school or junior high school: seventh grade through ninth grade
High school (or senior high): tenth grade through twelfth grade (or sophomore, junior, senior)
(common, but decreasing)
Elementary school: Kindergarden, first grade through eighth grade
High school: ninth grade through twelfth grade (or freshman, sophomore, junior, senior)
(extremely rare; only in very small and isolated schools)
Kindergarden through twelfth grade (possibly even in a single classroom - the so-called One-Room Schoolhouse)
(I once attended a school in which all grades from kindergarden through eighth grade were combined in two rooms with two teachers; there were nineteen students.)
To speak of all these grades collectively, sometimes the abbreviations "El-Hi" or "K-12" are used. (This is educator-speak, and not much used by others.)
I would say "He is in the second grade," "He is a second-grader," or "He is in elementary school." I would not combine the two: "He is in the second grade in/of elementary school."
(When giving just the grade, the reader/listener is expected to understand what school is implied. Where there is ambiguity {7th - 9th} the listener should ask, if interested: "Is that in a Junior-high system?")
I would say "He is a junior," "He is an eleventh-grader," or "He is in the eleventh grade." I might also say "He is in high school," or "He is a junior in high school," but not "He is in the eleventh grade in/of high school."
Note that the named grades (freshman, etc.) are the only ones that can be combined with the type of school. (I suspect that this is to avoid confusion with the similarly-named "grades" in college.)
Class is generally only used to describe the year of graduation from a school: "He is in the Class of 2013."
(Formerly only used in reference to college and high school, but recently used for middle school and even elementary school.)
First class would only be used to describe the quality of a school. "I am sending my child to a first-class high school."