This is such an incredibly broad question that the answer to which could take me hours to explain. If possible, could you please narrow it down between certain sections of the bible. Example: The book's of Moses (Genesis, Exodus, Leviticus, Numbers, Deuteronomy) or The Gospels or the Letters of Paul.

If you are looking for an answer to the bible as a whole, there are three books I would most likely recommend you start with since a brief explanation on Reddit would not really be able to cover these topics accurately.

So, two things real quick. The first, is that I’ve actually answered this question in different ways before (it’s also under the FAQ part on r/askhistorians as well) but I’d be happy to still answer it, but keep in mind I will be copying and pasting much of my answers from previous posts. Second, I want to throw in a disclaimer about the content of my response. I will answer it from the historical (not the religious or cultural) perspective of the known history of the historical ancient Israel. Much of what is in the bible and/or Torah cannot be verified or backed up using historical/archeological evidence. This still opens up much for people to learn since most of what we know about Israel's history is actually quite different from what we know in the bible. So in short, I am saying: although the answer to your question is quite different from what you have in the bible, it is not intended to be an attack anyone's beliefs or faith. However, I will try and make a correlation between what the bible says and what we know historically about the Israelites. (I say this because I have been down voted before when I said things which conflict with the bible and I don't want that to happen here.)

Primary Sources on Ancient Israel

So first off, it’s important to know that we have absolutely no original manuscripts of the Torah in any complete form dating before the first millennia BCE. This makes it difficult to know the authenticity of the claims in the book of Exodus since most of the claims should have happened hundreds or thousands of years earlier.

Likewise, there aren’t many ancient accounts from other nations about ancient Israel from before the first millennia BCE with the earliest being an Egyptian Stele which dates to just before the year 0. World Renown Egyptian historian Eric Cline wrote about the very first mention by anyone of Israel and its people outside of the bible.

Merneptah is perhaps best known to students of the ancient Near East as the Egyptian pharaoh who first uses the term “Israel,” in an inscription dating to this same year (1207 BC). This inscription is the earliest occurrence of the name Israel outside the Bible. In the Pharaonic inscription, the name—written with a special sign to indicate that it is a people rather than just a place— appears in a brief description of a campaign to the region of Canaan, where the people whom he calls “Israel” were located. Source: Cline, Eric H. (2014-03-23). 1177 B.C.: The Year Civilization Collapsed (Turning Points in Ancient History) Princeton University Press. (pp. 6-7)

What’s my point for all this: we have very little evidence about anything in ancient Israel’s past from before the year 1200 BCE. Even more important is that there isn’t any mention of ancient Israelites traveling outside of the Levant (the geographical area where Israel is today) in any great numbers, let alone being enslaved by them.

Likely Origins of Ancient Israel

So it's difficult to say what exactly was happening with the ancient Israelites. We have no archeological evidence for the existence of Moses or Joseph or even Abraham.

The Bible Unearthed: Archaeology's New Vision of Ancient Israel and the Origin of Sacred Texts by Israel Finkelstein. This is a great popular history book written by a well-respected historian which very plainly explains that the likelihood of the ancient Israelites to have been slaves in Egypt, be freed and walked across the peninsula to Canan and conquer it is extremely unlikely.

Rather, these historians explain that Israel likely originated in this area and the exodus story evolved from tales of the Hyksos migration from Egypt years prior.

The Exodus

So if the Exodus was true, when could it have happened?

The Exodus would have had to have happened after the year 1280 BCE at its very earliest (again, no credible historians believe this is the case). How do we know this? Well, part of why is that we have to match up the descriptions in the bible with known history.

If we look the book of Exodus, chapter 12:37 it says, "The Israelites journeyed from Rameses to Sukkoth. There were about six hundred thousand men on foot, besides women and children." Now I'm more concerned with the first sentence where it says they left Rameses for Sukkoth.

Now Rameses was the ancient capital of the 19th Dynasty of Ancient Egyptian history and founded by it's king Ramses II. The city, known in it's full name as Pi-Ramesses, was founded at some point during his reign (which was from about 1280-12:20 BCE). Now there is a small chance that this site could have been founded under his father, Rameses I, but even that would have put the founding of the city at no earlier than 1290. My point for why this is significant is because if the bible asserts that this is the city the fleeing Hebrew slaves left from, then there isn't a way that the book of Exodus could have happened any earlier than this time period. On the flip side, if we focus purely on the historical evidence that we have, it would suggest that the tribe of Israel were currently or about to become a vassal to the Egyptian thrown (as I said above). This area, known as the Levant, was almost exclusively under Egyptian control which was not necessarily a bad thing since the the Hitite kingdom (in the north near present day Turkey) were in a phase of expansion at the beginning of the 13th century BCE and it was beneficial for anyone's survival to be under Egyptian control rather than to be on your own.

Now another issue with the Exodus story is the sheer scale they mention. Again, let’s look at the Torah for their description: “At the end of the 430 years, to the very day, all the LORD's divisions left Egypt." Exodus 12:41 And they say that at least 600,000 men, plus women and children all left at at the same time in a great migration (Exodus 12:31). You’re talking about anywhere from 1 to 2 million people traveling across a pretty big distance for 40 years without any archeological evidence that was left behind them (not to mention that there is no Egyptian evidence that they enslaved any group or nation for this long). Biblical Archeologist William G. Dever wrote an interesting article for the peer-reviewed journal The American Schools of Oriental Research entitled Ceramics, Ethnicity, and the Question of Israel's Origins and he focused a lot on Israel's early history and how we know what we know today, and he explains in detail the evidence that we have of ancient Israel and how it doesn’t match up with the biblical story of the Exodus. If you can't locate it yourself, I'll try and see if I can locate it somewhere on the internet for free if you need this. (William G. Dever, The Biblical Archaeologist, Vol. 58, No. 4, Pots & People (Dec., 1995), pp. 200-213)

TL;DR Summary

Many historians believe that one reason how the Exodus story came about was because during the late bronze Age (which is this period of around the year 1200 BCE), Egypt controlled all these areas around the Levant. Israel was likely a vassal or was occupied by Egypt, and the Israelites hated them for it and these stories in the Torah are likely a combination of folk stories mixed in with rare elements of truth.

Hope this helps. If you have any follow up questions, please let me know.

*edits for grammar, clarity, and spelling