The modern term 'Religion' seems to be incorrect in a Roman setting. The term PIETAS doesn't seem to mean piety, as we know it; it seems to be more a reverence for, and a strict sense of duty or justice towards, the gods; although that didn't stop the Romans cheating on them. Religion meant more a binding obligation with its source a fear of the unknown.

Cicero (De Natura Deorum II.iii.8 - 9) could, despite being a sceptic on certain aspects of religious practice, still state "If we care to compare our national characteristics with those of foreign peoples, we shall find that, while in all other respects we are only the equals or even the inferiors of others, yet in the sense of religion, that is in reverence for the gods, we are far superior". This preoccupation with correct religious practices is also confirmed by the Greek historian Polybius who claimed that it "Is actually the element which holds the Roman state together" (Histories VI, 56.2)

Inevitably our image of Roman religion is obscured by the limited nature of the available evidence. Almost all of it comes from the Italy which gives rise to the question "How Roman is our view of the subject?"

There were wide variations in terms of the patron deities, the festivals and the rituals used throughout Italy as well as variations over time. What rituals were being used in the provinces must be the sticky question to be asked, and probably one that's impossible to answer.

The religions of Roman times started out, in early Etruscan times, as a set of rustic beliefs that later on got overlaid with a much more sophisticated set from Greece. Roman and Greek gods and goddesses became conflated until it was virtually impossible to separate them. Later still there were a number of 'imported' beliefs from Egypt, Palestine, Syria etc.

Clearly anthropomorphic deities such as the Etruscan trinity of Juno, Jupiter and Minerva spoke well with the Greek deities such as Juno = Hera, Jupiter = Zeus and Minerva = Pallas and became conflated with them, therefore it was inevitable that the Romans took the Greek ideas' to clothe their own austere set of deities. By the same token it was also inevitable that Oriental cults got included, albeit reluctantly, into the pantheon; thus came Bacchus (Dionysus), Cybele and Isis. However, these did not come to corrupt a race of pious puritans. They had been beaten to it, the native gods such as Fortuna, Virilis, Flora and Anna Perenna had a licentiousness all their own. We should be aware that the simple rustic is seldom a puritan, blood sacrifices such as on Fordicidia, when a cow in calf was killed, were common and human sacrifice, although extremely rare [illegal after 97 B.C.], was not unheard of.

Although this is a gross over simplification of a complex set of ideas', the relationship between man and god was more one of a business contract: 'You do this for me and I'll do that for you'. A man may even venture to defraud the god in question!

A universal state demands a universal religion!

Many of the 'official' pantheon offered by the Palatine Trinity, had neither pageantry or religious ceremonial, nor the hope of either immortality or spiritual progress. Some foreign cults - Greek and Oriental - imported during the late Republic and early Empire did just that and fulfilled the basic needs.
Amongst the most successful were 'The Mystery Cults' whose membership was confined to the initiates. Mithraism was arguably one of the more successful of these, although its popular appeal was curtailed by the Roman insistence [it didn't start out that way] on the limitation of membership to high-ranking males. It would seem that the Romans were never what you may call a religious people, but they were ritualists and formalists to the core.

Thus the religious practices were always conservative.

If for example the Fratres Arvales only jumped four times instead of five when intoning, chanting or singing the last line their sacred hymn then the whole value of the rite would be lost. This, at best, meant that they'd have to start all over again. Understandably enough some of the festivals and forms of worship seem to have echoes in Christian worship.

There was a definition of Public as against Private Religion, although there was a tendency for the two to run together, in the 'De Significatione Verborum' of Sextus Pompeius Festus (2nd C.) This implies that Public religion is that which is carried out at public expense either on behalf of the populus as a whole or of specific subsections of it, although private individuals could be permitted to finance temples and festivals on behalf of the People. Private religion is on behalf of individuals, familiae and gentes as well as certain other small groups.
Private religion didn't require the services of state priests, but there were several areas of overlap between the two and they weren't independent entities. Both public and private religion were officially overseen by the state through the ius divinum, however, public religion was restricted to the worship of a limited number of approved gods and select festivals whilst private religion - providing that there wasn't a breach of the peace - allowed for the worship of any god and the celebration of any festival.
Thus state pontifices would give advice on private religious matters and on other matters that may concern the individuals standing within the community.
Legally there was also a major difference between things dedicated to the gods as part of public and private religion; things dedicated for the people became consecrated Res Sacra and the theft of such items would constitute Sacrilegium, and hence a possible death sentence, whereas private dedications did not.

Arval Brethren. A priestly order who celebrated the Ops or Dea Dia festivals with a ritual, hopping, dance in 3:4 time.

Some of the rites that seem to have impinged on both Private and Public religion would be:-