The first-line recommendations I would make are the chapters on Anatolia in Mario Liverani's The Ancient Near East: History, Society, and Economy and Trevor Byrce's survey of Hittite political history The Kingdom of Hittites, the latter of which is generally considered the best and most up-to-date history of the Hittites around. Veenhof's Mesopotamia: The Old Assyrian Period is good historical background for the rise of the dynasty of Kussara and the prehistory of the Hittites(although they weren't in Mesopotamia proper, one of our major sources on Anatolia before the Hittites is the Assyrian trading colony at Kanesh and its records). If you are willing to go to the trouble of procuring academic books, H. Craig Melchert's The Luwians is probably a good overview of an oft-neglected aspect of Hittite studies. Unfortunately, most of the literature on Hittite culture is scattered around series volumes, individual essays and specialist publications(with the exception of Billie Jean Collins, The Hittites and Their World, and Bryce's somewhat iffy Life and Society in the Hittite World) and most of it is in Italian or German. Because of this, you may get more mileage consulting the this specialized bibilography of Hittites and Hittiteology. Of the numerous works presented there, probably the friendliest to a non-specialist are Itamar Singer's Hittite Myths. If you are looking for free online stuff, http://deepblue.lib.umich.edu/ has an extensive collection of free articles on Hittite-related topics by the prominent Hittite scholar Gary Beckman.

Anyhow, to answer your question, we actually know a surprising amount-much more than can be fitted into a reddit post, at any rate. By and large, we have a good handle on the outline of the political history of Hatti* and especially of the 'New Kingdom' rulers from Suppiluliuma I to Suppiluliuma II(a period which spanned 9 reigns counting those two monarchs) and excepting a brief period under the Old Kingdom rulers Hattusili I and Mursili I probably the pinnacle of Hittite political power. This and the diplomatic affairs of the Hittite kings, are probably the best-understood aspects of Hittite politics and are document both by the archives excavated at Hattusa and the archive of cuneiform diplomatic documents from the site of Armana in Egypt. Extending to this, our understanding of Hittite law and imperial administration is somewhat sketchy but in the process of improving dramatically thanks to new tablet finds.

We also have a somewhat detailed picture of the state religion and cult thanks to the extensive tablet archives found at Hattusa that give instructions for cultic actions and several mythological texts we've been very fortunate to find. We also have a Hittite law code, which has proved to be of great use in comparative studies of law in the Ancient Near East. Probably the biggest holes in our knowledge of the Hittites is life outside the government; the tablet archives at Hattusa are entirely from the government and until recent finds of provincial at Ortakoy and Masat Houyk they were pretty much our only textual sources on the Hittite empire-we don't have the extensive private archives and libraries you see for example in Babylonia. This means that our picture of Hittite life from below is pretty sketchy, relying on archaeology(and settlement archaeology is generally underdeveloped in the ancient Near East) and comparative linguistics. In addition, there are still some holes in the historical geography of Hittite Anatolia, although many holes have been filled in by the conclusive decipherment of the Hieroglyphic Luwian script and subsequent identification of a number of place-names formerly known only in cuneiform texts have been connected to specific geographical locations via royal monuments. Gary Beckman's article "From Hattusa to Carchemish" is a good overview of the more recent advances in Hittite scholarship(and of Hittite scholarship over the past century) and of the limits to our current historical knowledge.

*Hittite is a modern name derived from the Bible; to its inhabitants the Hittite kingdom was simply the "Land of Hatti" and its inhabitants were a bewildering range of ethnicities and speakers of an equally bewildering range of languages governed by a slowly shrinking ruling class of Nesite speakers. It is probably noteworthy that even though the administrative language of the empire remained Hittite until probably very close to the end, in later periods of Hittite history it was being throughly marginalized by the related Luwian language.

A good place to start is "The Hittites and their contemporaries in Asia Minor" by J. G. Macqueen. This book is a broad overview of the Hittite civilization and touches on warfare, society and its administration, daily life, religion, art and literature. It also has an extensive bibliography that will point you towards more in depth studies of the topics that really interest you.