Regarding grammar–it depends on what your ultimate goal is. Do you want to emulate the poor confused British schoolboys who were taught Latin grammar as applied to English (hint: it doesn’t really work), or would you like to study the state-of-the-art in syntactic theory?
If you want the state-of-the-art, then you want a red Cambridge book on Syntax, the most recent version of which was I think written by Andrew Radford. These books do assume that you know what the parts of speech are, and have some general idea that verbs take arguments, but are otherwise suitable for beginners.
If you want more of a historical standpoint, the reading is going to get heavier. You would start at the very beginning, with Noam Chomsky’s *Syntactic Structures * and Aspects of the Theory of Syntax. For reading past that, you would just be wading through oodles of tomes that are now considered obsolete, but that will ultimately lead up to the Radford you read in the preceding paragraph. To read all about this, look at the Wikipedia article on it.
Now. These theories are all general theories of syntax, and are supposed to be applicable to any language, anywhere. You’re going to want to know something about the rules of specific languages, and given that you’re posting on an English language message board and asking about classical education, my best guess is that you’ll want Greek, Latin, and English–although frankly, I think you could dispense with the Greek if you wanted. The contrast between Latin and English grammar is very strong, however. Latin has a strong case system, and English basically doesn’t have one at all. The standard text for learning Latin is *Wheelock’s Latin * and the accompanying workbook.
For English, any basic modern grammar would be good… to tell the truth, I don’t have a specific recommendation here because English is my native language, so I never took a class in its structure. If, however, you’d like to get down on Old English (which has a strong case system, like Latin), you’ll want Old English by Roger Lass. Then, if you want to read a bit about how the Old stuff turned into the Modern stuff, Millward’s *A Biography of the English Language * will explain it.
If you get through all this and understand it, you’ll know more syntax than the average linguistics Ph.D.
ETA: If you want to expand “grammar” to include all of linguistics, then the red Cambridge books are the way to go. They are all basic overviews of different areas of linguistics, written by people who are considered to be authorities in the area–though be aware that linguists are notoriously cliquish and unkind about approaches that are not theirs, so reading more than one overview of a subject area might be wise. The major areas of formal linguistics are phonology, morphology, syntax, and semantics. There are also sociolinguistics, anthropological linguistics, etc etc etc, but those are more sociological and anthropological studies.
