Whole books have been written about how to become a better writer, from the basic mechanics of constructing good sentences to marketing a finished novel. But there are three things that you absolutely must do to become a good writer:
Write a lot. Writing is just like any other discipline in that you won't get better at it just by thinking about it. The more you practice, the better you get at it. Try to write a little every day, in a journal, on a blog, or even just on the back of some scrap paper.
And don't just stick to one type of writing, either. If you're interested in fiction, for example, try writing poems and essays sometimes. By writing outside your "comfort zone," you can teach yourself something new about how the language works and how you can stretch it to suit your purpose.
Read a lot. There are plenty of books about writing and grammar and usage, and they can help you better understand how to write clearly and succinctly, but they can take you only so far. Learn how to write well from the people who are already good at it.
Find writers you love — novelists, poets, journalists, and even bloggers — and pay attention not only to what they say, but how they say it. Look at their word choice, sentence and paragraph length, pacing, and even punctuation. Then pay attention to those things in your own writing.
Reread everything you've written. It's a common saying among writers that "everybody writes crummy first drafts" (though crummy isn't usually the word of choice). First drafts often bear little resemblance to the finished product. Professional writers, though, have development editors, copy editors, and proofreaders to help turn that first draft into something worth reading. You don't have the luxury of an editorial staff, so you have to do it yourself.
You have to do it yourself.
Don't write something once and then call it done. Get into the habit of rereading what you write — not just stories and essays, but e-mails, blog comments, and even tweets — before you click Send or Print or Publish and call it done. This gives you the chance not only to find outright errors (and there will be errors) but to condense and clarify what you're trying to say.
As you find recurring problems in what you've already written, you'll soon find yourself paying more attention to those issues while you're writing, thus becoming a better writer.