Most of the money/resources schools receive comes in the form of grants.

Could you provide a source for that?

This claim definitely conflicts with my understanding, although perhaps it's true for that portion of resources that is actually up for grabs and not already committed through the normal funding (government) process.

This link is more in line with my understanding, that is, that most resources come from state and local government, and most of those resources are not awarded through "grants," but rather that local resources generally stay with local schools and state resources are divided in other ways but not usually through award of a grant. But I'd be interested in hearing if my (not heavily researched/sourced) understanding is incorrect either generally or at least for some portion of schools.

Some quotes from the link:

States rely primarily on income and sales taxes to fund elementary and secondary education. State legislatures generally determine the level and distribution of funding, following different rules and procedures depending on the state.

State funding for elementary and secondary education is generally distributed by formula. Many states use funding formulas that provide funding based on the number of pupils in a district. Some formulas are weighted based on different factors such as the number of students with disabilities, the number of students living in poverty, or the number of students for whom English is a second language. The allocation for students with different types of needs can vary significantly depending on the funding formula. Additionally, in some states the formula is designed so that higher poverty school districts with less access to local funding receive additional assistance.