The classic nuts and bolts guide to writing well is Strunk and White’s The Elements of Style. It strikes a great balance between teaching the mechanics and the spirit of good prose. Strunk and White cannot be avoided: it’s just too authoritative, and most good writers read it sooner or later.
But don’t start with it. Start with Zinsser.
The lesser-known but more readable and interesting book on English writing style is Zinsser’s On Writing Well. It’s less of a reference book, and more enjoyable. Although subtitled “The Classic Guide to Writing Nonfiction,” I think most of it is applicable to nearly any kind of writing.
There are lots of other great books on writing, of course, but this is my decisive short answer for the people who ask me.