Is "I know HERE good schools" correct too? Or is a difference if/when the important part of the sentence is "here" and not "good schools"?
It's just that English doesn't put an adverb after the verb, before the object ("know HERE good schools") like that.
Adverbs of place (like "here") tend to go at the very end of the clause.
Putting it at the beginning is possible ("Here, I know good schools"), but for this particular sentence I don't think it sounds very good.
Yes, exactly. I think it doesn't sound right because you basically put adverbs - especially ones dealing with location - at the beginning of a clause for purposes of contrast with another clause. (i.e. an implied one with the opposite adverb, e.g. Here, I know good schools. [Implied: But back where we used to live, the school system was a mess] )