Expand your vocabulary. Often people make tautological statements because they are using two imprecise words to describe something and don't have the vocabulary to express themselves in fewer words. One good way to expand your vocabulary is by reading books, magazines and websites that are written at an advanced level.
Think about the meanings of the words you use. For example, it may be obvious to you that a "mystery" is unexplained by definition, yet many people pair the word with "unsolved." "Unsolved mystery" is an unnecessary repetition because "unsolved" is already implied by using the word "mystery."
Look over a list of the most common tautologies. These include "free gift," "digital download" and "short summary." If making tautological statements is a common problem for you, studying lists of frequently used examples before you begin writing will put you in the mindset to aim for accuracy.
Do some practice exercises. One good way of learning to think of what words mean is by doing analogies. You can find practice analogies on the web or on practice guides for the GRE and SAT, two exams that rely heavily on these exercises. Analogies ask you to relate one word to another based on the precise definitions of the words. If you do many of these exercises, you will start thinking of words more precisely.
Revise your writing. Because many tautological statements are ingrained in the way our society uses language -- "Unsolved Mysteries" is the name of a popular television show -- avoiding them may not come naturally. It is very important, therefore, to read back your writing while specifically looking for these statements.
Limit your use of descriptive language in the first draft of your work. You can always add descriptions later if necessary. This is a more drastic approach if you find yourself continuing to make these mistakes. Most tautological statements are formed when a modifier is added to the subject, like in "new innovation." Write a draft focused on the bare bones information first. Then go back and add descriptions after asking yourself if your descriptions add unnecessarily repetitive information.