Read. Read. Read. Then read some more. If you want to write for a modern audience then read as many great works of fiction from the last 50 or 100 years as you can. Read the non-fiction too. Read great essays. Read The New Yorker. Then try this...

You know how when you go to an art museum you always see art students sitting around sketching some great masterpiece? Do that. When you read a great passage--or hell just open up something by a great author and pick a page--copy it down, word for word, and pay attention while you do it. Now go back through with a different color pen and break it down. How long are the sentences? Does their length vary? How? What letter starts each sentence? Is there alliteration? How is it stressed? And so on.

Reading will give you the tools, but like Alex said, you've got to write to get good at writing. Here's what I suggest. Either blog or write a journal every day. Then make it a habit to go back to things you've written (blog or work stuff, whatever) and review it, then revise it. Do that about one or two weeks after you've written it so you have fresh eyes. You'll start to pick up on your common mistakes and learn to correct and avoid them.

Above all else, a great writer must develop empathy for the reader. Become sensitive to every miscue or ambiguity in your writing. Empathize with the reader. Learn to express yourself simply and clearly.