Why did Jesus turn the water to wine (besides the fact the people had ran out) and why was it his first miracle?

The miracle to which you refer is recorded in John 2:1-11 [1] And the third day there was a marriage in Cana of Galilee; and the mother of Jesus was there: [2] And both Jesus was called, and his disciples, to the marriage. [3] And when they wanted wine, the mother of Jesus saith unto him, They have no wine. [4] Jesus saith unto her, Woman, what have I to do with thee? mine hour is not yet come. [5] His mother saith unto the servants, Whatsoever he saith unto you, do it. [6] And there were set there six waterpots of stone, after the manner of the purifying of the Jews, containing two or three firkins apiece. [7] Jesus saith unto them, Fill the waterpots with water. And they filled them up to the brim. [8] And he saith unto them, Draw out now, and bear unto the governor of the feast. And they bare it. [9] When the ruler of the feast had tasted the water that was made wine, and knew not whence it was: (but the servants which drew the water knew;) the governor of the feast called the bridegroom, [10] And saith unto him, Every man at the beginning doth set forth good wine; and when men have well drunk, then that which is worse: but thou hast kept the good wine until now. [11] This beginning of miracles did Jesus in Cana of Galilee, and manifested forth his glory; and his disciples believed on him.
See All.... As you mention, it is the first earthly miracle of Jesus Christ. The basic purpose of this miracle is given in John 2:11This beginning of miracles did Jesus in Cana of Galilee, and manifested forth his glory; and his disciples believed on him.
See All..., 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 1:14And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.
See All...) 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 2:22And 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.
See All..., 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 2:10And saith unto him, Every man at the beginning doth set forth good wine; and when men have well drunk, then that which is worse: but thou hast kept the good wine until now.
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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 27:28Therefore God give thee of the dew of heaven, and the fatness of the earth, and plenty of corn and wine:
See All...; Deuteronomy 7:13And he will love thee, and bless thee, and multiply thee: he will also bless the fruit of thy womb, and the fruit of thy land, thy corn, and thy wine, and thine oil, the increase of thy kine, and the flocks of thy sheep, in the land which he sware unto thy fathers to give thee.
See All...; Proverbs 3:10So shall thy barns be filled with plenty, and thy presses shall burst out with new wine.
See All...; Isaiah 65:8Thus saith the LORD, As the new wine is found in the cluster, and one saith, Destroy it not; for a blessing is in it: so will I do for my servants' sakes, that I may not destroy them all.
See All...; Amos 9:13Behold, the days come, saith the LORD, that the plowman shall overtake the reaper, and the treader of grapes him that soweth seed; and the mountains shall drop sweet wine, and all the hills shall melt.
See All...), joy (Psalm 4:7Thou hast put gladness in my heart, more than in the time that their corn and their wine increased.
See All...; 104:15
See All...; Ecclesiastes 9:7Go thy way, eat thy bread with joy, and drink thy wine with a merry heart; for God now accepteth thy works.
See All...; Jeremiah 48:33And joy and gladness is taken from the plentiful field, and from the land of Moab, and I have caused wine to fail from the winepresses: none shall tread with shouting; their shouting shall be no shouting.
See All...; Zechariah 9:17For how great is his goodness, and how great is his beauty! corn shall make the young men cheerful, and new wine the maids.
See All...; 10:7
See All...), and wisdom (Proverbs 9:1-6 [1] Wisdom hath builded her house, she hath hewn out her seven pillars: [2] She hath killed her beasts; she hath mingled her wine; she hath also furnished her table. [3] She hath sent forth her maidens: she crieth upon the highest places of the city, [4] Whoso is simple, let him turn in hither: as for him that wanteth understanding, she saith to him, [5] Come, eat of my bread, and drink of the wine which I have mingled. [6] Forsake the foolish, and live; and go in the way of understanding.
See All...). 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 36:26A new heart also will I give you, and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh.
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The water pots were to be filled with water. Water is a picture of the word of God (Ephesians 5:26That he might sanctify and cleanse it with the washing of water by the word,
See All...) and of the Spirit of God (John 7:38-39 [38] He that believeth on me, as the scripture hath said, out of his belly shall flow rivers of living water. [39] (But this spake he of the Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified.)
See All...). 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 [16] Jesus answered them, and said, My doctrine is not mine, but his that sent me. [17] If any man will do his will, he shall know of the doctrine, whether it be of God, or whether I speak of myself.
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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."