Your question is a little confusing. I take it to mean you are wondering about competing religions during the time of the Bible's compilation? Religions the people in the generations after Jesus would have encountered? Rather than in the time of Abraham and beyond?

First thing you'd have to understand is the Roman Empire was a multi-ethnic empire. There were a lot of religions flourishing within it. And the Romans, themselves, not only didn't mind but experimented with a lot of different cults. There is no negative implication of cult just a religious activity centered around one god or figure, in particular, although usually it didn't involve denying other gods. The only problems Romans had is when people wouldn't also sacrifice to the Roman gods or be willing to participate in some feasts. It wasn't about religion it was about loyalty to the state. This didn't come up much except with the Christians and Jews.

One very popular cult around the first century AD was the Mithraic Mysteries. Mithra is thought to have been an ancient Persian god. We don't know a lot about the mythology surrounding him. But it involved his defeating some sort of supernatural bull and then feasting on the bull with some other gods. The services were secret. Not because it was outlawed to be part of it (it was very popular with the army who would enforce those laws) but it was just part of the religion (perhaps to emphasize personal commune rather than state function). Christians seemed to especially hate Mithra because it seems like their rituals were copying Christian rituals. They had a bath of sorts, there was a marking on the forehead, the feast was an important part of it (like the bread and wine). St. Justin the Martyr (a very important early Church figure) wrote about his distaste for this. I think it's very unlikely that they actually stole from the Christians, though.

Another big one was the cult of Sol Invictus. This is important because it became an official cult of the Roman Emperors and used on their coins. Including Constantine the Great (who converted to Christianity on his death and was active in the Church long before). Sol Invictus is worship of the sun god. I don't know of any early Christian talk about it. But it was favored of the emperors even during conversion so they would have to tread lightly. Many believe a lot of the metaphors for Jesus come from this cult (the idea of Jesus as the sun, metaphorically, is something you see). There is some evidence that the major celebration for Sol Invictus was in late December and one calendar that has the date as December 25. Significant, obviously, due to Christmas. Whether Christmas is actually celebrated on December 25 because of this is hotly debated. I have no idea.

The cult of Cybele was very important in Rome going as far back as the Second war against Carthage. But also very controversial. She was a very ancient Mother goddess. The most famous part of this cult is the initiates of the priesthood had to castrate themselves. This horrified but also fascinated the Romans. Their chief priests would be comprised of high-ranking citizens (who didn't castrate themselves). St. Augustine writes about them in disgust he found the emasculated priests effeminate and disgraceful.

The cult of Isis was very popular as well. Including with Roman emperors like Caligula and Vespasian. She was the Egyptian mother goddesses. She was the wife of Osiris who she resurrected after he was murdered and disembodied by Set. She was also the mother of Horus and much of her imagery, especially in Rome, is of her nursing the baby Horus. This is thought to have inspired some of the imagery around Mary.

Several of the Emperors and Julius Caesar also had cults devoted to them. The senate could deify them after their death but some parts of the Empire (in the east) would worship them while they were alive. Augustus and his wife Livia (eventually), Claudius, Hadrian, Trajan, Marcus Aurelius, and many others achieved deification after their death. Not all of them were equally popular in worship. Caesar and Augustus were always popular ones. Hadrian was a popular god. There are temples to various defied emperors all through out the empire.

A popular religion in the Parthian and then the official religion of the Persian empire at the time was Zoroastrianism. It wasn't that popular within the Roman empire. This is a very ancient religion and Jews and Christians would have come across it in some form. I can't even begin to do it justice but it's significant for the similarities with Christianity. They believed the Wise Lord was the only, universal god, the beginning and the end, the Truth. They had concepts of heaven and hell. The believed active participation in life and doing good deeds was necessary (which is also an important concept in early Christianity and current day Catholicism and not as common in religions as you might think).

Today there is a lot of Christian pushback against the early Christians borrowing ideas from other religions. But I see it only as a strength. The underlying important part doesn't change (Jesus sacrificing himself). But why not be inclusive to the degree you can? Why not let people continue their feasts and celebrations? Why not use the compelling rhetoric? It includes people with absolutely no sacrifice of basic dogma.

Sources: Emperor Worship and Roman Religion

The World of Late Antiquity

A good book if you are looking for contemporary sources is Paganism and Christianity (which has a lot of the sources).

Pagans & Christians goes into more detail about the Pagan religions around at the time of Christianity's birth.

The novel Julian has a kind of interesting and well researched look into the connections. It is about the Emperor Julian and his attempt to return the Empire back to paganism. It does have a rather negative view toward Christianity so you may not like that part of it but it's otherwise very readable.