Abraham was before all of this, by at least several hundred years before Moses.
Abraham was a Bronze Age individual and his existence is really more legend than hard fact. One might think of him as being from 1800BC or as late as 1500 BC.
There seems to be some connection between the Hyksos and the migration of Joseph and his brothers into Egypt. The Hyksos are a fascinating subject in their own right. It is possible that the people of Abraham were the Hyksos themselves, or part of the general migration. The Hyksos probably arrived in Egypt around 1700BC to 1600BC. So that was the Abrahamites (Joseph, his brothers and their father Jacob, plus any others dependent upon them) opening to take the road to Egypt, possibly motivated by famine at home and/or the opportunity for cheap good well irrigated farmland.
The Hyksos were Semites, and controlled Canaan as far north as the whole coast of Lebanon, and possibly as far as the area around Antioch. They also controlled lower (but not upper) Egypt, where they set their capital. They settled primarily in the east Nile delta. You guessed it: Goshen.
Theoretically, if the Abrahamites were not the Hyksos themselves, they could have been Semites who migrated into Egypt in the wake of the conquest.
Eventually, the Hyksos dynasty was overthrown, and the Semitic population in the delta region would have been an unwelcome minority. Hence, the heavy taxation, corvee labor, confiscation of property and even imprisonment that they could expect as a conquered people associated with former oppressers.
When the Hyksos lost power, their whole territory was taken by the new Egyptian dynasty, all the way up the Mediterranean coast. The Bible gives us the idea that the Egyptians controlled the main road through Gaza and Megiddo to points north. The rough highland areas were not patrolled or effectively governed. Canaanite city states each had a ruler, called a king, who would be a vassal or ally of the Pharaoh. This might have been the model the Hyksos used to govern the region before.
While in Egypt, before the arrival of Moses to lead them, they would have been polytheists with merely a vague recollection of the religion of their father Abraham. When the Hebrews escaped the delta region, following Moses, they avoided the main trade route. This may have been roughly 1200BC. After this (and the entry into the highlands of Canaan) the tribes were in the period of Judges. At this point, I think we can assume the Hebrews were henotheists. They did not utterly destroy the Canaanites, they had to coexist with them. Apparently Jerusalem was not taken until King David’s time, possibly as much as two hundred years later.
Solomon’s temple (skipping ahead a bit) was erected in about the tenth century BC, initiating the First Temple Period. Interestingly, it was built by Canaanites in the style of a Canaanite temple (Phoenicians were Canaanites). Quite probably the Hebrews were still henotheists for the most part. The “high places” (local rugged temples, places of sacrifice) had been banned and destroyed, some priestly families were allotted places to serve in turn at Jerusalem. The prophets railed against the Hebrews for their faithlessness (more or less continually) for the next several hundred years. After Solomon died, the united kingdom of Israel split. Judah and Benjamin in the south to be known as Judah and Israel in the north. Israel allowed two more temples initially, so the religion had three officially tolerated sites of worship (plus numerous renegade temple sites, which may have been prone to allow worship of Baal and other Canaanite gods as well as Elohim). Judah never accepted the two new temples as legitimate, some people of Israel would come south to Jerusalem to worship, but could have also visited the Israelite temples back home.