Edit: by "writing skills" I meant general writing skills and not just technical writing.
Readable and fun, but extremely useful:
• Lapsing into a Comma, Bill Walsh (RIP)
• The 10% Solution, Ken Rand (for editing down wordy text)
• Sin and Syntax, Constance Hale
• Woe Is I, Patricia T. O'Conner
More resource-like and grammar-heavy:
• Words Into Type, Marjorie E. Skillin (hard-core grammar rules, great section on redundancies)
• Chicago Manual of Style (in general, the core style of book and magazine writing)
• Associated Press Stylebook (in general, the core style of newspaper writing)
• Garner's Modern English Usage, Bryan A. Garner
• Merriam-Webster's Dictionary of English Usage
• Dictionary of American Idioms, Maxine Tull Makkai
Not about writing, but hilarious and snobby and awesome -- great airplane reading:
• The Big Book of Beastly Mispronunciations, Charles Harrington Elster
Edit: One of the things he does really well is explaining the nuts and bolts "rules" of writing, like the Primacy/recency effect, the importance of what he calls "crunchy specificity", and lots of other stuff that will stay with you and help you make stylistic decisions in many different contexts.
2. Eats, Shoots & Leaves by Lynne Truss
3. The Elements of Style by Strunk & White
4. The Sense of Style by Steven Pinker
5. The Writing Life by Annie Dillard
6. Writing Tools by Roy Peter Clark
7. Bird by Bird by Anne Lamott
8. The Art of Plain Talk by Rudolf Flesch
9. A Writer's Reference by Diana Hacker
10. Writing Science by Joshua Schimel
If you get every punctuation mark right but don't have a single worthwhile thing to say, nobody is going to care that you're technically proficient. Nobody will even notice; they'll just move on to a less boring text. On the other hand, you can tell a perfectly good story even with a ton of misspelled words and malformed sentences. For example, the old novel Flowers for Algernon has large parts written in that style.
To that end, I don't think reading a bunch of technical manuals on "proper" English is anywhere near as helpful as just reading a lot of good books (browse bestseller lists, or classic literature, or Goodreads lists, and find a genre that appeals to you or a writer whose prose you enjoy). Then, to further your own writing, you just HAVE to keep writing and writing and writing AND get feedback. Take a writer's workshop course (medium or larger towns usually have a bunch; check your local newspaper or bookstore corkboard), or take a creative writing course at a local community college. No amount of self-study beats having twenty other people from various backgrounds give you feedback on your writing, specifically. It is horrifying to be so vulnerable, but it's the only way you'll know how your work resonates (or not).
Joseph M. Williams, Style: Ten Lessons in Clarity and Grace.
An old edition is fine. There are many editions with slightly differing titles (Toward Clarity and Grace, The Basics of Clarity and Grace), all of them are fine. Get the cheapest or fastest to deliver or whatever. Don't think about which one to get.