Use fewer damn words.
If I had to boil everything I know about how to write down into one rule, that'd be it. I was reminded of this by a post in Michelle Golden's Golden Practices blog.
I see redundancies in my work when I rush and almost always in a first draft or braindump. Calling myself out on it more often, I really notice it now in other people's writing.
There are some first drafts in which you've already written the thing in your head. Those tend to be fairly concise. There are, however, also many first drafts in which you're just thinking out loud. These tend to be absolutely crammed with unnecessary verbiage. It's especially bad if you dictate, which I sometimes do.
In these situations, a halfway decent job requires that you go back through your stuff, look at every single word, and keep asking "Do I need this?" Does the word contribute to the effect of the sentence, move your argument forward, or clarify something? If not, cut it.
You may have to repeat this process several times to remove all the chaff. But boy, does it pay off. Your writing will be way, way better, it will have much more impact on the reader, and you will train yourself, over time, to write more simply.
Winston Churchill understood this. I have always loved his description of how he learned to write simply and well as a schoolboy:
By being so long in the lowest form [at Harrow] I gained an immense advantage over the cleverer boys. . . . I got into my bones the essential structure of the ordinary British sentence - which is a noble thing.
You can become an outstanding writer by knowing what not to write.