Someone said without a source on another forum that for example Norse gods were seen as Roman gods with different names. (Just an example, idk if there was any contact at the time between Norse people and Rome, or even if Norse-identifying people or gods existed in 100BC.) How did society react to the newly captured religious influences, and how did the great writers of the time consider the holy men of what could be described as pagan cultures? How were foreign gods thought of by Romans?
This is a contentious question among historians and archaeologists of the Roman period, and I'm not even going to attempt to cover all the facets of the debate in one post. I'll suggest some reading at the end if you want to explore further.
In general, the Roman authorities were ambivalent to the worship of non-Roman gods. Their system was polytheistic (had many gods), so could easily accommodate new deities as new people were incorporated into the empire. There seems to have been a tendency to identify foreign gods with existing Roman ones, something Tacitus (Germania 43) calls interpretatio romana. Something similar must have been behind this section of Caesar’s Gallic Wars (6.17):
"They worship as their divinity, Mercury in particular, and have many images of him, and regard him as the inventor of all arts, they consider him the guide of their journeys and marches, and believe him to have great influence over the acquisition of gain and mercantile transactions. Next to him they worship Apollo, and Mars, and Jupiter, and Minerva; respecting these deities they have for the most part the same belief as other nations: that Apollo averts diseases, that Minerva imparts the invention of manufactures, that Jupiter possesses the sovereignty of the heavenly powers; that Mars presides over war."
From inscriptions and statues we know this really did impact on forms of worship in the provinces - local gods were given Roman names, or were symbolically married to Roman divine partners. Mercury and Rosmerta were the most common of these. Exactly how this interpretation worked is the controversial bit. Was it a top-down reorganisation by the Roman state? Or was it something the locals were happy to do to access the Roman power structures? It's a thorny issue, and it tied up with how we view Roman imperialism.
The only aspects of pre-Roman religious worship that the Romans were not keen on was human sacrifice. This was considered totally at odds with how Romans were supposed to behave, so they stamped it out quickly in the places they conquered. Tacitus paints a vivid picture of the destruction of Drudic shrines on the island of Anglesey:
"reassured by their general, and inciting each other never to flinch before a band of females and fanatics, they charged behind the standards, cut down all who met them, and enveloped the enemy in his own flames. The next step was to install a garrison among the conquered population, and to demolish the groves consecrated to their savage cults: for they considered it a duty to consult their deities by means of human entrails."
It seems that the druids were persecuted by the Romans, partly because of their possible engagement in human sacrifice but also because they might have formed some kind of political resistance to Roman rule. That's a good TLDR if there was one for this post: the Romans were mostly fine with other forms of worship, as long as it didn't directly go against their own sensibilities or threaten their political or military dominance.
Further reading as promised:
M. Henig 1984 Religion in Roman Britain
T. Derks 1998. Gods, Temples and Ritual Practices
Webster, Jane. "Creolizing the Roman Provinces." American Journal of Archaeology 105, no. 2 (2001): 209-25.
Webster, Jane. "At the End of the World: Druidic and Other Revitalization Movements in Post-Conquest Gaul and Britain." Britannia 30 (1999): 1-20.