Many writing guides provide endless quantities of style tips. It is easy to be pedantic and require that certain rules always be followed. But in reality, just about every rule (even spelling rules!) can be broken under the right circumstances. Even so, reading a good style guide is a great way to start thinking about how to improve essays at the level of words, phrases, or sentences. Experience must do the rest.
An easy beginning is the popular "Strunk and White" style guide; to take the next step, try the book by Joseph Williams called Style. As you read each book, look for principles rather than a simple list of rules.
There are two general ways to improve your style. The quick way, which is also quite effective, is to read your essay out loud or to listen while someone else reads it aloud. Awkward sentences will become painfully clear, and they can be fixed in the same day. The harder way, although it can have better results in the long run, is to imitate the style of an accomplished writer, as we did in imitating Tocqueville above, until you can write in that style naturally.