Read more books. It seems simple enough, but try to read as actively as possible. Try to see opposing points of views in stories and arguments. For example, if you're reading a work of fiction, try to see the story from the perspective of the hero, the villain and other major characters. You can also apply this to nonfiction. Simply pick a topic you're interested in and research arguments on both sides of the topic.
Study mathematics. Calculus, algebra and other mathematical subjects are based on logic and analysis. Learning to solve various math problems will build your analytical skills.
Solve puzzles. Work through logic puzzle books and look for puzzles in scientific magazines and on the Web.