The most important thing to understand when considering ancient religion is that a person of an ancient society would've considered religion entirely different than someone from a modern perspective. While this is fundamentally true for everything regarding antiquity, it's particularly true as regards systems of belief and ritual. Gould has a fairly recent article on understanding Greek religion, and of course there's always Nilsson's monumental work on Greek religion, Greek piety (mind you, it's very difficult to follow if you don't already have some clear understanding of the subject matter already). Greek and early Roman religion is where I'm strongest, so I can't really help much in the way of Near Eastern traditions, an once we enter the Principate stuff starts to change with the evolution of a different way of looking at things.
The point is that religious traditions are a fundamental cultural indicator, since they represent the most basic systems of cultural practice. Do away with any idea that doctrine or belief is necessary for religion. That's primarily a modern perspective, the result of centuries of intellectual development. On the most basic level, religious traditions are simply rituals. These are rituals and traditions specific to a certain group of people, which identifies them as separate from other groups just as certainly as language does, and it reinforces the unique codes of behavior within that society. This is most clearly seen in the language that is used to describe religion. Many cultures describe or described religious traditions as being the traditions of their ancestors or fathers--the Romans called it mos maiorum, the Greeks (depending where you were) had similar ways of describing it, and of course there are more obvious things like systems of ancestor worship within some cultures. In this way it becomes clear that funerary traditions occupy a crucial role in this hierarchy or ritual. A funerary rite must be performed with meticulous care--after all, this is a ritual in the manner of the ancestors that is being actively performed on an ancestor, so you'd better get it right. Because of this funerary traditions become important cultural thermometers, and picking them apart bit by bit can yield some very important results in the study of the fundamental values of that culture.
There's also another importance of funerary rites. Since funerary traditions are so deeply important within the cultural framework of ritual, they are very slow to change. Archaeologically, therefore, they become excellent tools not merely for dating (since a sharp break in the funerary rites of one culture can be used to date events easily) but also in identifying migration patterns or the relationship between cultures and peoples. For example, in addition to the purely linguistic work being done on Proto-Indo-European and its origins, there is archaeological work being done, attempting to identify common or closely related cultural patterns, which would indicate more closely the origin of what used to be called the fabled "Indo-European homeland," a concept now frightfully out of date. In migration especially funerary rituals change much more slowly than other cultural indicators (such as housing or clothing) and are closely on par with language. Additionally, funerary rites can also indicate either a shared (related) culture, or the spread of cultural influences between different groups. The adoptation of a funerary rite from a neighboring society is often an important indicator of deep cultural influence, since such things do not change easily (sometimes they can even be indicative of an actual takeover). For example, the great Urnfield cultures of Central Europe is one of the primary indicators of cultural spreading and influence throughout the region, since the practice probably originated somewhere in the northern Balkans and spread rapidly throughout the area (this is a heavily debated topic, since, particularly in northern Italy, there is a great deal of uncertainty just exactly how this ritual was spread and what the significance of it was. Nevertheless, it's a pretty good example).
I hope that gives you at least the beginning of an answer. To really fully understand the nature of the subject that you've opened up I highly recommend you take a look at the works I've listed and I'll see if I can't scare up any more.
For practical reasons mostly. Early on, I read "Gods, Graves and Scholars" a popular book for early archeology. I surmised that initially and to this day, grave sites provide the best insight into art, how people viewed their worth in life/afterlife, social hierarchies, and private life (artifacts buried with them.) Moreover, religion was the system of control for early societies - so funerary evidence is very helpful in this area.
In most cultures, these mark the few areas left untouched -- and even then there were grave robbers. The cities, written documents, and other aspects of day-to-day life were lost to the centuries. Ruins, vague folk memories, undecipherable carvings.
Some cultures have surviving written records - most notably the clay tablets of Mesopotamia, Crete and Archaic Greece and the papyrus of Egypt. These have to be deciphered and there is one system of writing from Crete (Minoan - Linear A) that I believe has not yet been.
Lately, with underwater archeology - we are finding shipwrecks that expand on the knowledge of ancient cultures. This shows what goods were traded, and in some cases, show surprising levels of advancement. It also gives insight into the commercial side of things, less likely under control of religion. Here is some information on Black Sea digs --