I've read that before one attempts to learn Latin it is best to study English grammar. Is that correct? If so, is there a recommended book on the subject?

I don't know about that. I do know that my English grammar is much better for having learned Latin, perhaps because I was taught Latin by a retired English teacher who taught English when English was really taught.

What about punctuation? I never see any...

Punctuation is a medieval development.

It wouldn't hurt to have a good grasp of English grammar, but I don't think it's necessary. In fact, I think that the explanations of grammatical concepts you'll find make a bit more sense for Latin than for English. And most good Latin grammars will get you started without too much trouble.

Wasn't Latin as well?

No. Spanish, Italian, and French were--all degenerate Latin, each of them.

With the admixture of a lot of distinctive vocabulary, and a good deal of grammatical simplification in all cases. Pronunciation also changed over time, with a lot of consonant clusters in Latin being leveled to doubled consonants or other such things in the Romance languages- Latin factum > Italian fatto and so on.

Who you callin' fatso?

Why learn the grammar of a different language?

Try Greek (pretty closer than English) or Finnish (15 different cases for a noun!).

Finnish Cases