1. Read good books. And magazines. And websites. Find people whose writing style is similar to what you would like to achieve, and read as much of their work as you can stomach. Read whatever interests you in a topic that you care about, and try to pay attention to how other people write about it.
2. Write lots, and lots, and lots of stuff. The exact topic and format are less important than choosing something you will take at least fairly seriously. Hell, start a blog.
3. Find someone that is willing to edit at least some of what you write, and willing to do so mercilessly. Not a close friend; find someone that can honestly evaluate your work. Listen to what they say. Look for the mistakes that you make the most frequently.
Other generally good advice is to read books about writing (which will depend on exactly what you're looking to accomplish). Proofread your own work (print it on paper and hold it upside down; it forces you to read slower and more carefully), have someone else look at it, and after some time passes, proofread it again. You'll find yourself coming up with better ways to phrase a lot of things.
Speeches work on a little different skill set, but the same ground rules apply. Listening to other people, getting in some practice, and seeking out criticism are all important.