To teach K-3, considered elementary, you need to major in elementary education. For 4-8, you major in elementary education and generally pick a subject or to to specialize in, such a math or language arts. This is because most schools have kids begin to switch classes at that time. In high school you major in secondary education and choose one area of specialty, such as math, science, etc. Special education teachers at elementary level major in elementary ed with a special ed endorsement, or go through a special education major program. If you teach high school, you do the special education program and have to be 'highly qualified' in any subject you teach the kids, i.e. resource English.
Each state has learning standards that every student must know. Each grade level has a different set of performance objectives under each standard that is particular to that grade. As the grade levels go up, the standards stay the same, but the objectives build off the objectives of the previous year. Example: add and subtract one digit numbers with regrouping may be a second grade objective under the number sense and operations standard. The third grade objective would be assing and subtracting two and three digit numbers with and without regrouping, but would still be under the number sense and operations standard.