Some people might say, "I'm not interested in how to improve my writing skills; I write well already. I just want to know how to get published."
It all goes hand in hand. Many of the things to improve your skills put you in a position to get published. If you are writing well, you will more likely get noticed. It's not all that uncommon for people (editors and publishers) to contact you and ask you to write for them. That's how I started writing for several years on assignment for a particular publication. An editor was told about something I'd done, and out of the blue that editor phoned me. So, don't allow your writing to be sloppy because you think, "This doesn't count--not that many people will see it."
Look around for books on writing. There are more great books on writing than can be counted. Writers Digest publishes a myriad of excellent books on writing.
If you don't want to buy your own books, check your local library. If they don't have a writing book you want, they can usually get it for you through a system called Interlibrary Loan. Because I underline, highlight, and make notes in my books, I have to buy them. I also keep them to use for reference books. By studying and applying what you learn, these writing books can vastly speed up the learning process. There are writing books for every single area of writing interest, including: humor, column writing, mystery and suspense, romance, personal essays, writing query letters, and more. There are books on individual techniques such as plot, structure, dialogue, conflict, etc.
There are often terrific sales on these writing books for people who like to buy their own books.
Places like http://www.halfpricebooks.com/ also offer books at less than full price. Another way to own your books for very little cost is through http://www.paperbackswap.com. On this site you can search for books you want and list books you are willing to give away. You then connect either with a person who has the book you want and can send to you or wants the book you have and you mail it to that person. The only cost is for postage in mailing. Amazon.com also lists varying prices for a book title.
However, you do need to buy, don't borrow, a good grammar book for the main language for which you are submitting to publishers. Even American English and British English are dissimilar in small idiosyncrasies so you need to take into account where you are seeking initial publication. You may never become the best grammarian but if you want to be taken seriously as a writer, grammar and punctuation are things you must learn to do better---maybe not 100% proficient at---but you must learn to be acceptable. If you do not have the grammar abilities you need, it's an absolute necessity that you find someone who can help edit your work before it is sent to editors.
Email is a simple way to focus on writing skills. Make sure you are saying things in the best way possible. Re-read your emails to verify your grammar and punctuation before hitting the send button.
It may be a new concept, but strive for vivid nouns and active verbs. With the possible exception of technical writing, creating a strong visual image through the use of vivid nouns and active verbs is very important in published writing.
Don't feel overwhelmed. Find one or two things to concentrate on first. Let that be your current writing focus before moving on to something else.