The Cuneiform Digital Library Initiative (CDLI) has a huge collection of scanned tablets. There you can see them in their vivid, beautiful splendor, but unless you learn to read Akkadian or Sumerian, the site will be of limited use to you.

Our reading list has a good summary of introductory texts on the Ancient Near East. All of these works include references to specific texts and many include translations. The Oxford Handbook of Cuneiform Culture, in particular, will give an introduction to the different genres and periods.

If you just want to read some stuff that was written in cuneiform writing, check out the Electronic Text Corpus of Sumerian Literature. They have translations of most of the important Sumerian stories there.

The best general introduction is probably Cuneiform by C.B.F. Walker. It'll tell you what's out there, how much of it, and where to find it.

/u/WedgeHead, /u/Xschmyesh, and /u/farquier are all correct. CDLI and Oracc are both the most well-known online databases for cuneiform tablets. A large number of tablets on CDLI (such as this 2nd millennium BC one from the city of Larsa) have beautiful photographs, and most also have bibliographical information, if you want to read more about the tablets. As /u/WedgeHead and /u/Xschmyesh pointed out, however, these tablets are not translated, and are not placed in any historical context. For the non-specialist, CDLI is of little use. Oracc is similar, but for some of the texts of some of the projects, there are at least translations. Also, contrary to what /u/farquier said, the projects cover a large number of periods, not just the late (i.e. 1st millennium) period. For instance, the Corpus of Kassite Sumerian Texts contains 2nd millennium texts from Babylonia, and the Electronic Text Corpus of Sumerian Royal Inscriptions has editions of Early Dynastic (3rd millennium BC) tablets. In the interest of full disclosure, I am on the staff of one of the Oracc projects, so I know a bit more about its intricacies. But, like CDLI, Oracc is aimed toward scholars, so it is very difficult for a non-specialist to navigate the site.

This is actually a problem that I have with Assyriology as a whole. Because of the difficulty in learning, reading, and thus translating ancient cuneiform languages, the documents are not very accessible to the layperson. Because of this, the texts have taken on a sort of mystic aura for some people (I refuse to link any of those sites, but you can easily Google them). Scholars and universities, I think, have not made a conscious effort to make these ancient documents more accessible to the public. Projects like CDLI and Oracc, while important for their own purposes, do no serve the wider general public.

If you're still with me, /u/madwilliamflint, then let me point you to one more site that may be of more help to you. Livius.org is an under-utilized but generally great site for accessing ancient documents and history. The site is maintained by a collection of scholars, and nearly all of the texts on the site are translated in English. Granted, the site only has a small percentage of the total number of cuneiform documents still extant, but the site has the major ones. For instance, it has a section on the Great Flood myth, and the Epic of Gilgamesh that is associated with it. It also has an English translation of the famed Cyrus Cylinder, a royal document of the founder of the Persian Empire. The site is also very in good in providing a historical context for many of the documents it includes.

If you have access to a library or don't mind purchasing something, I highly recommend J. B. Pritchard's The Ancient Near East: An Anthology of Texts and Pictures as well as Michael Roaf's Cultural Atlas of Mesopotamia and the Ancient Near East. Pritchard's volume contains English translation of all of the major cuneiform tablets from the ancient Near East, as well as short explanations of them. Roaf's book has beautiful pictures and maps of the ancient kingdoms, cities, statues, and cuneiform tablets from antiquity. I highly recommend both of them.

There is such a project, the Cuneiform Digital Library Initiative. However since the database is designed for researchers and not for general use, I would recommend to start this page on search terms

They also have a wiki and an open access journal!

A substantial corpus of cuneiform texts in translation is viewable via the Open Richly Annotated Cuneiform Corpus(http://oracc.museum.upenn.edu/); note however that not all periods are represented equally(the site seems to have stronger coverage in later periods of Cuneiform literature and texts) and not all sites provide translations or editions of all the works listed. But you can at least get a taste of some of of the kinds of texts Assyriologists read and work with.