I'm not too fussed here about specific time periods (Akkadian vs Third Dynasty of Ur, etc.) I'm just curious about what we actually know about these peoples. The stuff I've heard mentions them being nomadic, using hit-and-run tactics when at war with the Akkadians, and I think I heard they may have been pastoralists. Do we know anything about religion or social organisation? Are there any comparisons that can be made with later Turkic nomadic people? Did the Gutians have horses?
I'm sad to say that almost nothing is known about the Gutians. Calling them "nomadic" is not necessarily accurate; sources situate them in the Zagros mountains, so in terms of lifestyle they probably would not have resembled steppe pastoralists, but we don't really know in what manner they lived. In the third millenium there do not seem to have been horses in Mesopotamia in significant numbers, though there were donkeys or onagers that were also used to pull a type of war wagon depicted in the Standard of Ur. While the horse begun to be imported to Mesopotamia at the end of the third millenium, there is no attested connection to the Gutians; if the Gutians who destroyed the Akkadian Empire around 2100 BCE and then briefly ruled much of southern Mesopotamia had been a horse-based people, we might expect that their period of rule would have caused a more radical shift in the use of horses in Mesopotamia, which however happened only a few centuries later with the introduction of war chariots.
There's really no reliable way to relate the Gutians to any modern language or cultural group, since the Gutian language in the late third millennium is known only through very few personal names, and the written sources that mention the Gutians are consistently biased yet not very detailed, calling Gutians "brutish" and emphasising their lack of unified kingship as well as their transgressions against the gods and cults of Sumer. That said, the Gutian ruler of Nippur, Erridupizir, left a brief royal inscription in Sumerian, which indicates that the new rulers at least tried to take over pre-existing power structures and associated royal modes of presentation. So while the perception of Gutians as ravaging barbarians is probably at least partially justified (they did destroy the Akkadian Empire), the Sumerian sources also probably reflect a tendency to contrast the civilised Sumerians with the barbaric foreign Gutians, and aren't strictly reliable as sources about Gutian culture or way of life.
In short, we just don't know. All we can really say is that the Gutians left a major impression; the Mesopotamians seem to have seriously hated the Gutians, and in later times the very term "Gutian" became a kind of catch-all term for uncivilised mountain people without any real connection to ethnicity, not unlike the way Germans were called "Huns" during the world wars.