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I am on mobile, so apologies for the non-exhaustive references.
While I've never read either Dante or Milton (and therefore can't speak to that aspect) and can't speak to what "many Christians believe" beyond saying that that does match my personal experience, I can shed some light on what the (Protestant) Bible says regarding Satan. And while those who know more are welcome to chime in, I doubt that on this issue Catholic/Orthodox/Jewish scriptures have much different content, except perhaps in Satan's "origin story" and fall from Heaven.
It might be surprising to the Biblical newcomer to learn that Satan just doesn't show up in the Bible very much at all, but let's go through the most salient verses about him.
First, note that the Old Testament mentions Satan, but never mentions "devil" or "devils", though there are a few mentions of "demons" in Leviticus, Deuteronomy, and Psalms. But all three mentions of demons seem to treat "demon" as meaning "idol" or "false god" - they're all mentioned in the context of condemning those who sacrifice to them. Note that Leviticus does say "goat demons", which might have something to do with the modern image of the devil, which otherwise is very lacking in biblical basis (the Bible all but completely lacks physical descriptions of Satan / the Devil).
Most people are familiar with the serpent in the Garden of Eden who convinces Eve to break God's rule and eat from the forbidden fruit. But while many people sort of assume this is Satan, nothing in the Bible really indicates this, beyond some character similarities between the Eden serpent and, say, Satan the accuser's role in Job or during Jesus' temptation (and some Revelation imagery involving dragons and serpents). In fact, Genesis 3 seems rather focused on the serpent being a serpent, as his temptation of Eve is stated to be the reason why serpents have no legs - God removed them as punishment for his actions.
Job is the main book where Satan proper shows up, which is really quite interesting. Job is very much an outlier book. Most of the Old Testament is concerned with the history and fortunes of Israel and its people, and even most of the remainder (really, just a big part of Genesis) could be considered simply prelude to that. Job stands apart. None of the characters seem to be Israelites or directly related to Israelites. The best I can do to connect Job to Israel is to say the Abram's (Abraham's) original home is generally considered to be somewhere to the east of Israel, and so is Uz, Job's land of residence.
However, the Septuagint does claim he lives in Edom (a nation/region nearby Israel to the east) as a grandson of Esau, who was brother to Jacob (aka "Israel"). This would make him roughly a contemporary of some of the first Israelites born in Egypt (prior to the institution of the Levitical law) and is consistent with Lamentations 4:21, which seems to equate Uz and Edom:
Rejoice and be glad, O daughter of Edom, you who dwell in the land of Uz [...]
(Lamentations 4:21 ESV)
It's hard to even say what "religion" Job practices per se. He, his wife, and his friends clearly worship the same God as Israel and Jacob, and appear culturally similar to Israel, but there's no mention of Israel or of priests, though Job is explicitly stated to make sacrifices that are certainly at least reminiscent of those prescribed by Levitical law. One might conclude he engages in a form of proto-Judaism (like Noah or Abraham himself), or that he is a YHWH worshiper outside of Judaism (perhaps akin to Naaman of Syria), or that he does indeed participate in "The Law" of Israel, his sacrifices outside the prescribed procedures notwithstanding (since Israel itself did much the same thing throughout much of its Biblical history). This, among other things, has led some to argue that Job is not intended to be read literally, which poses a minor problem for using it as a basis on which to build an image of Satan.
In any case, Satan in the book of Job is at odds with God, and wishes to prove to God that those humans who are faithful to Him are only faithful because God blesses them. But he is also, so to speak, on speaking terms with God. He comes and goes before God (presumably in Heaven) and God gives him permission to test Job's faith - at first by material privation and deaths in his family, and eventually with disease and illness.
This "Satan the accuser" is elsewhere in the Old Testament. He is credited in 1 Chronicles 21 with inciting King David to conduct a census (considered a sin because the motivation was to assess Israel's military strength, which betrayed a lack of trust in God to win their battles for them). He is present in the vision in Zechariah 3, where an angel rebukes him and reassures Joshua "the high priest" that God will bless him if he is righteous (clearly intended as a general message to the Israelites, with Joshua [the man who presided over the initial conquests of Israel/Canaan] as a stand-in for Israel as a group). Satan is present "to accuse him".
And that's basically it for Old Testament mentions of Satan. Based on this source material, many argue that Old Testament Satan isn't even evil; that he is more analogous to a state prosecutor than an agent or architect of evil. But note that if Satan were a modern prosecutor, his cases might get thrown out on the basis of entrapment.
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You might find this answer by u/sunagainstgold useful. You can also check these answers by u/amayo20 , u/YCNH and u/fistantellmore that talk about the connection between Lucifer and the Devil