Reddit mentions of History Begins at Sumer: Thirty-Nine Firsts in Recorded History

I'm no expert on this subject, but History Begins at Sumer might help you out. From what I remember, the Sumerians did not have much in the way of chronicle or history, let alone historiography. So, the answer is, they probably weren't really aware of the past behind a few generations.

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
  
  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.
  
  
• ETCSL
  
  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
  
  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:
  
  
• History Begins at Sumer
  
• The Sumerians: Their History, Culture, and Character
  
  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.
  
  
• Religion in Ancient 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.
  
  
• The Treasures of Darkness: A History of Mesopotamian Religion
  
• The Harps That Once: Sumerian Poetry in Translation
  
  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.

I love this site. It's not "records" such as census records or tax or wage receipts, but poetry, myths, and a few letters. It's got translations as well as transliterations if you wanna see what Sumerian "sounded" like. Very cool, in my non-professional opinion.

I'd also suggest "History Begins at Sumer" by Samuel Noah Kramer. It's older; First edition was written in '56, and I've got the third edition which was published in '81- but I still found it a great introduction.