Mesopotamian mythology (including the Sumerians, Akkadians, Assyrians, and Babylonians) is actually one of the better documented mythologies of the ancient world. Despite the preponderance of "ancient alien" nonsense out there, a plethora of scholarly books exist on the subject, as well as websites curated by academic groups.
ORACC is an excellent place to start if you're interested in the major deities of Mesopotamia. Each entry contains historical information (cult centers, earliest attestation, etymology, etc), and well as mythology (genealogical, function in pantheon, etc). The entries are sourced, so you can follow up any queries you may have.
The ETCSL is a collection of Mesopotamian mythology in easy-to-digest format. Most of the material on here was written during the Old Babylonian period (c. 1800 BCE) but is based on earlier traditions found in the Akkadian and Ur-III periods (c. 2300-2000 BCE). A print-version of many of the myths contained on the webpage can be found on Amazon here.
SEAL operates much the same as the ETCSL, only it is focused exclusively on literature from the Akkadian empire (c. 2350 BCE). If you follow the "corpus" link you'll be taken to the entire collection of texts SEAL has available for browsing.
There are other such collections, similar to ETCSL and SEAL that are available online, but these are good starting points if you don't yet have your feet wet regarding the subject matter. You should also, if you're interested, explore print-material as well. Much of Mesopotamia's core research is published in book format, by professors and archaeologists who're still working on dig-sites or in universities.
A few names and texts to consider:
Samuel Noah Kramer, who published the above, provides an excellent introduction to Sumerian culture and mythology. His books are easily readable and often times serve as most people's introductions to Mesopotamia.
Jean Bottéro's masterwork has been translated, but is approachable and easy to read. This volume explores the emotional and psychological connection of the Mesopotamian people to their religion.
Thorkild Jacobsen is one of the best translators of Mesopotamian material. These two books present dozens of myths, cult songs, and other works for the reader, while also exploring the patterns and motifs present in Mesopotamian religion across its 3000+ year history.
There are other works I could recommend, but I don't want to overwhelm you with options. If you're interested, I also run the subreddit /r/Sumer, which is dedicated to reconstructing the religious beliefs and practices of the Mesopotamian people. It isn't the most active of places, but I'm still trying to get it off the ground. Feel free to join us over there if you wish.
Check out Jeremy Black's The Literature of Ancient Sumer.
I recommend either: The Literature of Ancient Sumer or The Harp that sounds which is older but still good
Black, et. al's The Literature of Sumer has a nice collection of Sumerian Myth and B.R. Foster's Before the Muses has a wide collection of Sumerian and Akkadian-language myths. Both in translation.
The Electronic Text Corpus of Sumerian Literature (ETCSL) catalogs Sumerian language compositions from the Lagash-II, Ur-III, and Old Babylonian periods, ca. 2200-1595 BCE. When people talk about "Sumerian mythology" they are usually referring to texts from this corpus. The Literature of Ancient Sumer collects many of the ETCSL compositions together in print form, and The Harps That Once... provides translations of some additional texts not available on the ETCSL.
As Iskuss1418 suggested, you can also browse our reading list (available in the sidebar and About page) for a more thorough collection of books. The academic resources link (available in the same place) will take you to a list of online sources for texts, translations, analysis, and more as well.
You bring this subject up every few months, and never seem to follow any of the links that I continue to provide for you. If you're looking for mythology, kings lists, prayers and petitions, cultic songs and hymns, or any other kind of composition, it is already available:
Mesopotamian Polytheism Resource List
Mesopotamian Polytheism Reading List
Both of these lists have been freely available in the sidebar/About page of this community almost since its inception, right under the Rules and above the remaining theological resources.
If you explore the links, you'll find access to: the SEAL project, which provides translations of Old Akkadian Period literature, as well as publication information; the ETCSL, which provides a catalogue of Old Babylonian Period compositions; and the ETCSRI, RIAo, RIBo, and RINAP, which collect royal literature (inscriptions, historical compositions, regnal years, etc.) from Sumer, Akkad, Assyria, and Babylonia across Mesopotamia's lengthy history.
There are also genre-specific collections, like: the DCCLT for lexical material; the BTTo for topographical texts (temple lists, city lists, etc.); the DCCMT for mathematical literature; the ADsD for astronomical literature; the CMAwRo for material related to witchcraft and magic; the DSSt for scribal diatribes; the DLL for love lyrics; Dr. Alan Lenzi's Shuilla and miscellaneous prayer collections; and the AMGG for overviews of deities.
And that's just some of what the resource list I've compiled includes.
The reading list also includes some of the gold-standard books of the field, arranged by subject matter, like:
Just to point out a handful of the commonly suggested resources from the list.
As for financial considerations: I will never monetize anything in this community, never ask for donations, never charge anyone a membership fee, or put content behind a paywall to the best of my ability. My purpose in taking up responsibility for this community was to do the exact opposite: to make the material accessible, free, and available in language that both new devotees, and seasoned practitioners could appreciate.
To that end, there's also a very robust community, from the Facebook Temples, the Order of the Rod and Ring, The Four Reeds non-profit, The Ishtar Gate, and more (all available in the sidebar/About page) that assist in this goal.
The sidebar/About page already does this for the material that is currently available online.
There are also a handful of print publications that go a long way toward aiding in this process too:
The difficulty with doing a "weekly myth" is that there are, at conservative estimates, half a million cuneiform tablets, and that any given myth is a compilation of tablets, anywhere from 2 to 30+. So, as inexhaustible as my resources may seem, such a project is simply beyond my ability. Entire universities are dedicated to doing this with just a single time period or genre.
If you have questions about a particular text though, you're always welcome to ask and I'll do what I can to provide answers, or point you toward someone who can.