Anyone wanting to learn Akkadian should acquire a copy of Huehnergard's A Grammar of Akkadian. It provides a very comprehensive overview of Old Babylonian grammar, and the readings draw upon different genres of texts. The book comes with an answer key, which is essential for a self-learner.

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), consult 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. A good overview of the cultural context of cuneiform is found in Cuneiform by Finkel and Taylor.

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 good resource for the most important cuneiform signs is Snell's Workbook of Cuneiform Signs.

There are several other languages that used cuneiform, and the availability of grammars and dictionaries is extremely variable from one to the next. You'll have to be more specific if you want resources for moving beyond Akkadian.