John 1.35-42

³⁵ The next day John was there again with two of his disciples. ³⁶ When he saw Jesus passing by, he said, “Look, the Lamb of God!”
³⁷ When the two disciples heard him say this, they followed Jesus. ³⁸ Turning around, Jesus saw them following and asked, “What do you want?”
They said, “Rabbi” (which means “Teacher”), “where are you staying?”
³⁹ “Come,” he replied, “and you will see.”
So they went and saw where he was staying, and they spent that day with him. It was about four in the afternoon.
⁴⁰ Andrew, Simon Peter’s brother, was one of the two who heard what John had said and who had followed Jesus. ⁴¹ The first thing Andrew did was to find his brother Simon and tell him, “We have found the Messiah” (that is, the Christ). ⁴² And he brought him to Jesus.
Jesus looked at him and said, “You are Simon son of John. You will be called Cephas” (which, when translated, is Peter^[g]). John 1

It is perhaps important to discuss that, earlier in this book, the verb usage of the crowds of people and the people from the priests coming to John meant that they kept coming; coming and leaving and returning. And that this journey was not an easy one--a rocky terrain that had its ups and downs both coming and going.  (Mack)

John knew that he would lessen, his job was winding down as Jesus' job was taking off. He needed the people to turn and see Christ, not continue to see him. Think, if you will, of Paul writing to the Corinthians, thanking God that he had not baptized but a few men. The people were arguing of who they followed, who, as a teacher, was more right---when in all honesty none is "right" enough, good enough, or perfect enough but the one true Teacher, our Christ. 

2What I mean is this: One of you says, “I follow Paul”; another, “I follow Apollos”; another, “I follow Cephas^b ”; still another, “I follow Christ.”

13Is Christ divided? Was Paul crucified for you? Were you baptized in the name of Paul? 14I thank God that I did not baptize any of you except Crispus and Gaius, 15so no one can say that you were baptized in my name. 16(Yes, I also baptized the household of Stephanas; beyond that, I don’t remember if I baptized anyone else.) 17For Christ did not send me to baptize, but to preach the gospel—not with wisdom and eloquence, lest the cross of Christ be emptied of its power. 1 Cor 1 NIV

John's work would be defeated if the people continued to follow him, insisting that he was the teacher they expected. In fact, in Acts 19, Paul comes across some followers of John the Baptist who had not heard the Holy Spirit and were still going on what John had taught them but we're pleased to hear and receive the Spirit.

John, therefore, pointed out to his followers the Lamb of God. If John, whom they had known for at least some time, pointed out this man, a certain amount of curiosity had to be raised. Who was this man, that John said such thing of him? And, if John's loyal followers shift their attention over to another Man, this will bring attention (I assume) to the Man in question by the people coming and going from the river to hear John. After all, how fast does gossip, true or false, take to wander about a group of people to reach further and further circles? Especially of such a grand idea as a possible Messiah!

So off they go, to speak to this man. They honor Him with the name "rabbi", teacher---  "it was a word meaning "great one" or "my master" which was applied to many kinds of people in everyday speech... a measure of their great respect for him as a person and as a teacher and not just a reference to the activity of teaching he was engaged in." Any "advanced" teacher, such as Christ,  or John, would attract many disciples who were desperate to learn, as became more and more prevalent as teaching took off. ( Vander Laan )

Jesus asks what these men want, and they ask where He is staying---and HE INVITES them to go along. This, as I have learned, is a bigger deal than I have ever been told. Anyone might ask someone to coffee that they are enjoying speaking to. Some might invite new people to their home after church.This is no let's go have coffee sort of question as we will find out.

Asking where Christ was staying has a deeper meaning than we might think: "In the Jewish world, this question was the means by which a talmid [disciple] would submit himself to the teaching of a rabbi. If the rabbi essentially said it was none of his concern, then that person would be rejected as a talmid. But the opposite was also true. If the rabbi said, “Come and see,” that person would be accepted as his talmid." ( Mack )  As our story progresses we will also find the usual trait if Christ that He asks the followers to follow Him. Most rabbis had want-to-be students following them about to see if the teacher might accept them. Children applied to become disciples of rabbis of good standing. Few rabbis ever asked random people they passed by to become followers. (Vander Laan):

²⁷ After this, Jesus went out and saw a tax collector by the name of Levi sitting at his tax booth. “Follow me,” Jesus said to him, ²⁸ and Levi got up, left everything and followed him. Luke 5 NIV

  

⁴³ The next day Jesus decided to leave for Galilee. Finding Philip, he said to him, “Follow me.” John 1 NIV

Rabbis of any repute had disciples under them constantly. It was said by Jewish teachings " 'Let your house be a meeting place for the rabbis, and cover yourself in the dust of their feet, and drink in their words thirstily (Pirke Avot 1:4)'. The best talmidim [disciples] (plural) were the ones who stayed so close to their rabbi that they could take in every detail of their mentorship. That should be a fresh challenge today as we consider the call of Yeshua upon our lives."(Mack ) 
Further, the disciple was a "mirror" of the teacher, and as such, the teacher was a "mirror" of the disciple---puts a new perspective on if we act poorly, we reflect poorly on our Lord, though He Himself never acts poorly:

“The disciple is not above his teacher, nor the servant above his lord. It is enough for the disciple that he be like his teacher, and the servant like his lord.” Matt 10.24-25; .Moss

May the Lord help us to be like Him and to not reflect poorly on the One and Only God!

Another interesting point is that the disciple was to be able to represent his teacher in legal discussions, and the actions of the disciple were a reflection on the "family name" of the teacher, as they were considered a "household" with the rabbi. Disciples, lastly, were also students in whatever trade that the teacher was in, if he had a trade. Moss gives the example of a blacksmith rabbi who would then teach his followers to do as such.( Moss )

There could be a deeper learning by being with the Teacher at all times, seeing Him at nearly every moment of every day. Seeing a Man who was Perfection in and of Itself. Seeing a Man and how He knew the Word, teaching constantly of how the Word was, how to teach, how to minister, how to act. Seeing how He was with others spoke beyond measure to the disciples--of any teacher ever. He, who spoke to a Samaritan woman, a terribly shocking thing to do, and reached the town of people by this would have changed His disciples opinions and actions. John 4.1-42 ; Moss

Part of the teaching of a rabbi was to ensure that his students have the ideal opportunities to study and listen to him. ( Moss ). He also spoke seperately to the disciples to make certain that they understood what He had been saying. He allowed them to ask questions: 

 
 Now when Jesus saw the crowds, he went up on a mountainside and sat down. His disciples came to him, ² and he began to teach them. Matt 5 NIV

 10The disciples came to him and asked, “Why do you speak to the people in parables?” Matt 13

As we know, eventually, some people believed in Him, and some were merely entranced by the man who made miracles and puzzling statements. It was important for the closest followers to learn deeply, to then later, as we know they did, teach others the words He spoke. It was tremendously important for the different crowds to hear what Christ had to say also; we will look at teaching the crowds vs the disciples at a later entry--- 

³⁵ That day when evening came, he said to his disciples, “Let us go over to the other side.” ³⁶ Leaving the crowd behind, they took him along, just as he was, in the boat. There were also other boats with him. (Mark 4). 

And, as a good rabbi does with His talmid (disciples), when they have become like Him, He sends them out to the world to find more who would imitate them, and thus imitate Him. ( Matt 10 NIV  ;  Vander Laan    )

After listening to Jesus all day, Andrew rushes off to find his brother, for who else might one run to with such news as the Messiah but to one's own family member?

 Upon their return to Christ's side, it seems that no names were ever even said, but that He who made all knew Peter by sight, studied him with an intent, piercing gaze which reads the heart, according to the Greek, ( Mack )--and renamed him. An interesting section from Barnes' Notes on the Bible states (BibleHub):
"Cephas - This is a Syriac word, meaning the same as the Greek word Peter, a stone. See the notes at Matthew 16:17. The stone, or rock, is a symbol of firmness and steadiness of character - a trait in Peter's character after the ascension of Jesus that was very remarkable. before the death of Jesus he was rash, headlong, variable; and it is one proof of the omniscience of Jesus that he saw that Peter "would" possess a character that would be expressed appropriately by the word "stone" or "rock.""

So calming, really, to consider that this was what Christ thought of Peter even though he would deny Christ three times before His death, followed with Christ's post-resurrection questioning three times  if he loved Christ, as I think, a solidifying of their relationship.