The miracle to which you refer is recorded in John 2:1-11. As you mention, it is the first earthly miracle of Jesus Christ. The basic purpose of this miracle is given in John 2:11, which states, "This beginning of miracles did Jesus in Cana of Galilee, and manifested forth his glory; and his disciples believed on him." According to this verse, the miracle had two purposes. It proved the deity of Christ (John 1:14) and it strengthened the faith of the disciples. However, as you, I am intrigued as to why this particular work was God's choice for the first miracle of Christ.
I think much of it has to do with the typology of the miracle. Jesus uses "new wine" as a type of the new teaching and the new truth He brought to earth. In Mark 2:22, Jesus taught, "And no man putteth new wine into old bottles: else the new wine doth burst the bottles, and the wine is spilled, and the bottles will be marred: but new wine must be put into new bottles." His teaching is the new wine. The unscriptural institutions and traditions of the Judaism of His day are the old bottles. Jesus announces His new wine with His first miracle. It is better than the old wine (John 2:10).
In the Bible, wine is used as a type of many things, including wrath and judgment. However, this miracle clearly points to the positive connotations of wine. In scripture, wine is a type of blessing (Genesis 27:28; Deuteronomy 7:13; Proverbs 3:10; Isaiah 65:8; Amos 9:13), joy (Psalm 4:7; 104:15; Ecclesiastes 9:7; Jeremiah 48:33; Zechariah 9:17; 10:7), and wisdom (Proverbs 9:1-6). The new wine Jesus brings to Israel is all this and more.
We can see a more developed type in this miracle in the dispensational changes that are to come. Let us look at this type in depth. The setting for the miracle is a picture of dead Judaism or of any other form of dead religion. First of all, the wine ran out (v.2); they lost their joy and blessing. Then, Mary tried to take the lead. This reminds us of the emphasis of Catholicism on Mary.
The water pots were to provide for the purification of the Jews (v.6); this is a picture of religious ceremony. But they were empty; as are the empty ceremonies of man. There were six of them: the number of man (as man was created on the sixth day). They were made of stone; like their stony hearts (Ezekiel 36:26).
The water pots were to be filled with water. Water is a picture of the word of God (Ephesians 5:26) and of the Spirit of God (John 7:38-39). This is the starting place for any revival. They were filled to the brim. ( A firkin was about 9 gallons. That made the total about 120 gallons or so.) By filling the water pots to the brim, they were emptied of everything else - just as we need to be emptied of self in order to be filled with the Spirit. Finally, the water was drawn out and taken to the guests. We must take the truth of God to others. But only the servants knew the full extent of the miracle (v.9). By application, those who do the will of the Father know the doctrine (John 7:16-17).
Jesus brought the new wine of His truth to us. We need to take this new wine to the world. As Jesus told the disciples after His resurrection, "Peace be unto you: as my Father hath sent me, even so send I you."