Anyone wanting to learn Akkadian should use Huehnergard's A Grammar of Akkadian (free PDF courtesy of the author). It provides a very comprehensive overview of Old Babylonian grammar, and the readings draw upon different genres of texts. If you're teaching yourself, be sure to acquire a copy of the answer key as well.

An Akkadian Handbook: Helps, Paradigms, Glossary, Logograms, and Sign List by Miller and Shipp is a very useful volume that I still consult from time to time. The paradigm charts are particularly helpful for a beginner.

The standard dictionary for students is A Concise Dictionary of Akkadian by Black, Postgate, and George. The Chicago Assyrian Dictionary (CAD) and Von Soden's Akkadisches Handwörterbuch are more comprehensive but not as useful for looking up words quickly.

One of the hardest parts of learning Akkadian is the memorization of cuneiform signs. Cuneiform began as a pictographic system like Egyptian hieroglyphs, but cuneiform signs became more abstract over time. Neo-Assyrian signs in particular can look quite different from the corresponding signs of the Akkadian or Ur III period. For a good overview of how signs change over time (including their origins), there's Labat's Manuel d'Epigraphie Akkadienne. Knowing the origins of the signs and how they were first written can help you remember them more easily.

The best way to learn cuneiform is to write out signs again and again. Make sure to go from the syllabic or logographic value to the cuneiform, not just the other way around. (In other words, be able to draw, say, a DINGIR/an or LÚ sign.) It's something that takes a lot of practice, so start with only a few signs a day. If you're using Huehnergard, he provides signs to memorize in each chapter. A great resource for the most important cuneiform signs is Snell's Workbook of Cuneiform Signs.

Finally, I recommend Cuneiform by Finkel and Taylor for an overview of the cultural context of cuneiform.