Every native speaker of every language "takes grammar seriously" in the sense that it has become wired into our brains from immersion and long practice.

People may not know the terminology of grammar, but almost nobody says "Tomorrow I gets up early" or "Yesterday I buyed foods on the supermarket." Both of those sentence are completely clear and understandable. Neither of them sounds like slang. But both of them sound wrong, instantly and obviously. 

When I was chatting with a Spanish language companion and I said "el quejó," she didn't say "quejarse is always a reflexive verb," she just said "You mean 'el se quejó.'" She knew the actual grammar. She took it seriously. 

There are different levels of formality, "registers;" some kinds of word choice and grammar signal "educated speech" or social status; almost everyone uses "better English" when writing than when speaking. It is better to think of informal (or uneducated English or "dialect") as having _different_ grammar, not as "not taking grammar seriously."

As a learning strategy, I personally think it is helpful to me, and accelerates my learning, to know some grammar. I think it is helpful to an English learner for someone to say, out loud, "usually, verbs add an 's' in the third person singular; 'I buy,' but 'he buys.' And 'buy' is irregular, and the past tense is 'bought.'" A native speaker who is not a teacher might never say a rule like that out loud, but I think 99% of native speakers follow that particular grammar rule 99% of the time.

If I am writing a cover letter for a job application I take grammar very seriously indeed. When I am editing a Wikipedia article, or posting a forum article like this one, I try to use "correct" grammar. 

I was taught grammar when I was at school as part of English Language classes.  However, in everyday life, as I grew up, I paid very little attention to grammar when I spoke.  That's because it was not necessary.  Natives will instinctively know what other natives are all talking about, even with bad grammar.  It does not mean that when necessary, in a formal situation we will not change and speak with near perfect grammar, it just means that we are more relaxed with grammar in everyday speech.