For a beginner, I would recommend "Complete Babylonian" (Teach Yourself) by Martin Worthington.
It is very accessible. After having some basic knowledge, there are more complete grammars and dictionaries.
The reason for learning an ancient language is reading the original texts. There is no need to become an expert, because there are bilingual editions for most ancient texts.
In my experience, no matter how good the translator is, you can find much more interesting information in the original text, but using a bilingual edition spares you 99% of the time you would have needed to read unaided when that is not your base profession.
For Akkadian, the best text for beginners is the bilingual edition of the code of laws of Hammurabi, which is very clear and easy to read (There is a better modern edition by M. E. J. Richardson, "Hammurabi's Laws", but also an 1904 edition at archive.org, "The Code of Hammurabi").
After that, the bilingual edition of Gilgamesh is recommended, with both the Old Babylonian version and with the later versions.
Unfortunately, the bilingual edition of Gilgamesh (A. R. George, "The Babylonian Gilgamesh Epic" in 2 volumes) is quite expensive (though there are pirated copies on the Web).
Thank you for the recommendations. That's very helpful. I'm less bullish on bilingual editions (I know Ancient Greek well, and Latin a little less well, so I'm not totally uninformed) but I'm unlikely to ever be as proficient at Akkadian as I am at the languages I studied in college so I'm glad to know those exist.
John Huehnergard's A Grammar of Akkadian: https://brill.com/view/title/38184 seems to be well-respected. It's a grammar textbook that goes through the usual topics (nouns, verbs, etc.) Teaches the dialect of the Old Babylonian Empire and should be approachable for someone who has studied another language before.
A History of the Akkadian Language: https://brill.com/view/title/21744 was just published. It's monumental and eclectic. All sorts of topics related to Akkadian. Its internal history, changes, as well as recent history of its deciphering. I've been skimming that, using that as a sort of backgrounder. Some of it's over my head, and I took a few semesters of linguistics once upon a time.
The Chicago Assyrian Dictionary has been fully published and digitized: https://en.wikipedia.org/wiki/Chicago_Assyrian_Dictionary But it's like the OED and I think at the start you would want a wordbook or wordlist with your reading materials, that matches the dialect/era you're working with. A variety of those floating around.
While the written form was pretty fossilized (like Latin or English spelling) it was used for ~3000 years and changed a great deal over time like any language. Old Babylonian seems to be a common starting point. I think probably because it's well-attested, was some of the first material discovered, and because it's the language of (the most complete version of) Gilgamesh, and also Hammurabi's law codes.
