I think this varies from place to place. I learned what was "correct" as a kid like most people do, namely listening to older people and reading a lot. They taught us spelling through fifth grade (so about age 10-11), although obviously not everyone needed it that long while others never got it...plenty of adults still can't spell for shit. But I think some of the problem, as others have alluded to, is that there's very different "versions" of English. I know you and I have talked about this before. I grew up middle class, white, and with well-educated parents, so what I heard from my family (and friends, who tended to come from similar backgrounds) was the more educated register, which is usually what someone means when they talk about "correct" English. For someone growing up in an inner city, at best they may've heard that kind of English on the news or something, but it's not what they're used to day-to-day. So in many respects they come to school having to almost learn a foreign language, but I don't think this fact gets quite enough recognition (or at least didn't when I was in school). Instead, they're told that what they and everyone they love is speaking is "wrong," and then we're surprised when they push back or refuse to learn it out of pique. This remains something we have to deal with: a huge part of my job is explaining complicated legal and medical concepts in ways that can be understood by someone who never made it out of 8th grade. But it would sound really condescending if I were to switch into, say, AAVE in a formal court decision, and that's without getting into the problem of how much profanity is an integral part of that way of speaking.
But for my part, I've never diagrammed a sentence in my life. English classes were mostly just learning by doing, whether it was reading or writing. Over the course of the next couple of years "English class" transitioned to be strictly about literature, which I also hated. It wasn't until I started learning foreign languages that I began learning linguistic terminology. But certainly by 9th grade (so age 14) our English classes were exclusively focused on reading "great" works and writing the occasional paper.
Thinking back to when I first started learning Russian, I really didn't find the grammar to be that hard. (My problem is that I suck at learning vocabulary, so that was always far harder.) But putting the pieces together correctly (with the correct endings and what-not) never really caused me any trouble. I actually came to enjoy Russian's pith, like the way you could use дательный падеж for so much, and how many incredibly versatile words Russian has (вот is one of my personal favorites). Random anecdote. I was in the dining hall one summer (taking summer classes), and the guy next to me was evidently Russian. He saw a cute girl across the way, and said under his breath: вот эта такая. That was the whole expression, and it expressed his feelings perfectly. But putting it into English is surprisingly difficult...we just don't have constructions like that. Now you're making me miss Russian :(