This is a longish chapter, but it is all logically and temporally contiguous. It is pretty apparent that it should be treated as a single unit, so that's probably the best way to read it.
There are a number of minor points to address and then some bigger points. First, some minor points. Second, I have heard some interesting teachings on this chapter and I will recap some of those analyses once the minor stuff is done.
The servant is not named in this chapter. This has resulted in some speculation as to his identity. The main identity assigned to him is Eliezer of Damascus, who is mentioned earlier as Abraham's chief servant and the inheritor of his estate, if he had no son. However, this claim is unconfirmed.
There are a lot of cultural differences between the events of this story and modern life. This is certainly not an exception as there are many differences throughout the entire OT as a whole. Obviously these characters are living in a time when arranged marriage is customary, and Abraham as the patriarchal authority, is responsible for bringing this about.
Another noteworthy fact is Abraham's insistence upon Isaac marrying a relative. This is also largely contrary to modern social rules, but is in keeping with Abraham's marriage to his half-sister; also, Nahor and Milkah are closely related. In modern society, we frown upon it for essentially genetic reasons.
However, I believe that the primary concern of Abraham and the people of his generation is spiritual contamination from the idolatrous peoples they live amongst. This is not actually stated, however: Abraham never says why he wants Isaac to marry amongst his people. He implies the rationale when discussing the Lord though: He does not want Isaac to marry a woman who might dissuade him from following the Lord, which would essentially invalidate the promises given to him.
Furthermore, as I previously discussed, marital separation and purity are strongly promoted concepts in the rest of the Pentateuch, as well as the OT in general, so I don't think it's unreasonable to say that there is some intended correlation here. Not necessarily that the writer of Genesis backfilled events, but at least as likely that the history of Abraham and Isaac informed their later opinions about marital separation from the peoples they lived near. As Abraham and Isaac are two of the most revered figures in Judaism, it should come as little surprise that their choices would guide the spiritual teachings of Israelites for many generations afterwards.
The biggest problem with this whole discourse is that as far as we know, Abraham's family does not actually worship the Lord either. To the best of my knowledge, when Abraham is called by the Lord back in chapter 12, that was essentially a divine intervention: neither Abraham nor anyone else in his family had knowledge of the Lord at that time. Upon further thought, this opens up a whole new window of interpreting that passage: perhaps God drew Abraham away from his family and land for the deliberate purpose of separating him from his past, which would have pulled him back into idolatry. If indeed his whole family and nation was idolatrous, then they would have precisely the same pull into godlessness that Abraham is trying to keep away from Isaac, and that Moses will later try to keep away from the children if Israel (without success, I might add).
So yeah, I'm not really sure what to conclude about Abraham's insistence here. It's probably related to keeping Isaac from idolatry, but I don't know how that fits in with Laban/Rebekah's spirituality. I guess the biggest factor might be a desire to maintain a national/racial identity, separate from the Canaanites'. This is related to worshiping the Lord, because the Lord essentially becomes the patron God of Abraham and his descendants, in the fashion of the time (again, it's a cultural thing). If their national identity were intermixed with the much more numerous Canaanites, they would almost definitely be absorbed into the local (idolatrous) culture.
The other strong insistence of Abraham is that Isaac should remain in Canaan. This one is much simpler and he explains it perfectly: God is going to give this land to his descendants, so his descendants should remain here and not go back.
Verse 50 implies they know of the Lord, but this doesn't necessarily mean they follow the Lord (they were likely told of him by Abraham before he left). It's also possible that, having observed the great wealth the unnamed servant brought with him, they were quite pleased to accept what bridal price he would offer, regardless of the story of divine fortune that he told them. Laban certainly proves himself later to be a very savvy and sometimes deceptive businessman.
The end of the story is, Rebekah comforts Isaac after his mother's death. This is a fitting note, because Isaac, being a sole child and the promised child given to Sarah after she was barren for so many decades, would obviously have been spoiled rotten. Anyone who knows what even a normal mother is like should know that Isaac was probably given anything he could want, so it's humorous to think of Abraham going to get a wife for Isaac so that he would have someone to continue taking care of him after his mother dies. This is not, perhaps, exactly how it happened, but it's funny to think about.
Another thing to note is the repetition of the story, when the servant gives his account. This is precisely the sort of repeated storytelling that implicates the underlying oral narrative that preceded the written Torah. As I was taught in high school, it is good form to repeat information when speaking, because people need to hear things multiple times to absorb them, but it is terrible form to write things multiple times, because anyone can just re-read a given segment as often as they need. It's interesting to note that, in general, these repetitions are largely contained in the Pentateuch, and not in later OT books, which gives an implied older date to the Pentateuch, which has a strong oral tradition compared to the more obviously written form of the later works. Or, as so many others believe, the JEDP theory is true but only applies to the Pentateuch.
Lastly, I will mention that it's worth considering the character of Rebekah in these passages. As others have pointed out, almost all of Rebekah's actions belie a very bold, daring personality. First, we see her water the servant's camels without even being asked. She offers to do it voluntarily. Second, we see her agree to travel with the servant literally the day after he arrives, and she leaves her family to marry someone she's never met. Again, arranged marriages were much more common back then, so this is not entirely surprising. But it's interesting to draw a parallel between Rebekah departing for a husband she doesn't know and Abraham departing for a land he does not know. Both of them seem to have a degree of fearlessness in their actions that shapes how they react to the circumstances presented to them.