Quite a bit, but you have to dig to find it.
In HoME 10 (Morgoth's Ring) there is the Athrabeth Finrod Ah Andreth, which includes the story of the original Fall of Men (which happened in the far eastern land of Hildorien - not included on any map), which some say happened during their first generation:
Then one came among us, in our own form visible, but greater and more beautiful...
You can probably guess that things go slightly (to say the least) downhill from there!
Interestingly, and if we accept this Tale (it is just a mythology within a mythology, after all) there is a chance that this was probably Sauron, not Melkor, based on the following evidence:
- According to the main chronology, Melkor would have just returned from killing the Trees in Valinor at that time, and very likely had other things on his mind; Sauron had been left behind during the Chaining of Melkor so he was effectively a free agent.
- The description of the "one" who "came among us" is totally consistent with Sauron's behaviour elsewhere in the Second Age.
- The subsequent human sacrifice, burning, building of a temple, etc, is also totally consistent with Sauron's behaviour elsewhere in the Second Age.
- Melkor's inability to change his form after his destruction of the Trees (by the main chronology the Awakening of Men was later than that, so Melkor - as Morgoth - could not have been "far more beautiful" at that time).
Here we don't see much of the culture of later worshippers, but we do learn what worshipping Sauron involves - as I said, lots of human sacrifice, lots of burning, building a big temple, and nice things like that. This is also seen in the Akallabeth (tale of the downfall of Numenor, published in the Silmarillion).
Later on, in HoME 12 (The Peoples of Middle Earth), we have the story of Tal-Elmar, which presents the coming of the Numenoreans from the perspective of the Wild Men of Middle Earth, again during the Second Age. This is an interesting piece with regard to your question, as it shows how the average run-of-the-mill/not-directly-involved-in-Sauron-worship Man views the world, and gives a decent idea of the kind of culture they live in.
Also in HoME 12 there is the essay "Of Dwarves and Men" which mainly focussed on "High Men" and "Middle Men", but also briefly mentions "Men of Darkness".
The Silmarillion itself is a source for certain information about the Easterlings, their early people, which sides they took in the Wars, etc. Some information is available in the Chapters 18 and 20.
LotR Appendix A is a major source for information about the Wainrider culture of the Easterlings, and further information is available in Unfinished Tales (Cirion and Eorl).
There is, so far as I am aware, nothing major about the Haradrim, although both Appendix A and Cirion and Eorl again provide some about the Corsairs of Umbar.