The best way to do this is to do the Matasano crypto challenges. The challenges are designed to get progressively more difficult, and since the goal of most of them is to break something, that means you'll learn how to employ progressively more sophisticated attacks. They're also great for a newbie, because in one challenge you'll implement something that seems impervious to attack, then in the following challenge you'll attack and break it, often via an unexpected attack vector. (The padding oracle attack comes to mind.)

You'll come away with an understanding of AES in ECB CBC and CTR modes; HMAC; timing attacks; attacks on RNGs; attacks on hashes; and a lot more. That's just off the top of my head.

For me, there were a dozen "aha!" moments in the first 30 challenges. Each of those moments now live with me and inform my future decisions. They will also make you much less confident in your ability to design secure cryptosystems, which is good.