It is important to keep in mind that there really is no such thing as "the" Roman religion. All attempt to reconstruct what "the Romans" believed are efforts to make a more or less cohesive whole out of something that was in all likelihood much more amorphous. I think what we so charmingly call "traditional Chionese religion" is a good copmpüarison, and if you read a book on that you will very soon lose any illusion of finding coherence or logic. Taking, e.g., the cult of the Capitoline Triad (Iupiter, Iuno and Minerva) that was pacticed throughout the Empire, it is hard to say what role this played in the lives of participants. Most likely, it functioned as an aspect of civic religion, something like Independence Day parades.
As was said before, the main thing about religious practice in the Roman tradition was that it was bound up with social and political structures. That does not mean, though,m that it was something like a state cult (it is too easy to draw parallels with the royal summepiscopate or vilayet-e-faqih). Cicero, who was a much better theologian than he is usually given credit for, observed that the Immortal Gods emerge from the actions of mortal men; that men worship the good things they do. And traditional Roman religion had a deity for pretty much everything - the fasti list them with tedious (and spurious) precision. That is also where our traditional idea of "God" gets in the way of understanding. THe Roman world was full of Gods - every home had lares, every family manes, every person a genius or iuno, every road crossing had lares compitales and every spring nymphae. To most religious Romans, an existence at peace and respectful distance from those beings was what 'religion' was all about. In rural context, it is likely that those practices continued far after they had become antiquarian oddities to educated urbanites.
On top of that was a layer of civic religion, practices that bound together communities. The Capitoline deities were shared by all Roman citizens, though not with equal devotion. The cult of the imperial genius (a nice touch, since only "bad" emperors demanded to be worshipped as Gods, but nobody could object to a cult of the emperor's guardian spirit) was originally a pretty limited affair, but spread. In addition to those, there were vocationally specific cults (e.g. the deities of the signa and the camp for soldiers, Isis among sailors) that are found in much of the Empire. Regional communities had shared cults, often with festivals people travelled very far to attend such as the feriae Latinae, the Olympic and Isthmian games, or the Eleusinian mysteries. And every city had a collection of shrines at which its citizens practiced religious ritual together. Being part of these activities cemented your communal status. In either case, BTW, there was no question of belief. Cicero believed that the Divine preexisted humanity, and that the contrary assumption was a mark of a bad character, but he himself had to admit that the Gods most likely were not real persons. The pax deorum did not require belief. If you participated in the rites that defined your community, you were a member of that community in good standing asnd could believe whatever you wanted.
In addition, there was private religious (or philosophical) practice. Philosophical schools would have been the closest, socially speaking, to what we think of as "religions": groups of people reading the same scripture, sharing the same beliefs about body, soul, and afterlife, a moral teaching, and a dogma that they debated and interpreted. A lot of the vocabulary of Christianity - othodoxy, dogma, heresy - comes from these schools. Philosopher-preachers could earn money in much the same way Evangelicals do now, by assembling a community of the faithful and collecting donations in return for guidance and spiritual comfort. Some of the abuses, too, appear to have been similar.
Together with philosophical schools, more traditiuonally religious cults became popular for individuals throughout the Empire. Many of them remain quite mysterious to us. Iuppiter Dolichenus was worshipped by individuals as far away as the Rhine frontier, and we would dearly wish to know how or why. Isis was extremely popular. And of course, synagogue-going became a common practice. I sometimes envision rabbis as something like Tibetan Buddhist lamas today: we don't really understand what they do, but surely they must be much more spiritual than us corrupted Westerners (apparently, Tibetan Buddhism has considerable pop appeal in China, too). All these things were firmly in the private sphere. The broader community only interfered when it felt that there were practices contrary to propriety or dangerous to the social order.
What an individual believed or practiced religiously depended on where that person located him- or herself geographically, socially and personally. Religion did not unify people in an overarching church or ummah. The religious calendar of a soldier would be a very different one from that of a farmer or a freedman barber. Calling all of this "a" religion is difficult, but of course, in Latin, "religio" is not properly countable. It's a property of persons and societies, not a specific practice or belief. That is why, as many Romans saw it, the Christians lacked "religio".