There are currently two in-game newspapers coded in the game, to which you can subscribe or simply purchase editions.
Once you have purchased an edition, if you are logged into the website, you can read it here. Editions not purchased are listed, but cannot be read until you purchase them in game.
The current edition of either newspaper can be bought at the appropriate newspaper office (or anywhere else there is a newspaper box), from the paperboys and papergirls who wander around Ankh-Morpork, or from Sneed Finkman. You can also subscribe to either paper to get new editions deposited for you at the post office. A subscription can be either pre-paid, for a particular number of editions, or unlimited.
Old editions can be bought in the newspaper offices, in the archive rooms. You can get either a paper edition, or a website edition. In both cases you will be able to read the paper on the website, but with the website edition you will not get the in-game physical item.
Despite their lack of a coded newspaper, that did not prevent djelians from making their own!
There are 14 editions of the papyrus available there: https://djbpapyrus.proboards.com/