Pentecost & the Ascension
First, of interest in passing, is that it is Jesus who pours out the Spirit on the day of Pentecost.
Secondly, what remained unfulfilled of Israel's OT exile was the re-establishment of a (legitimate) king, and the outpouring of the prophesied Spirit. The definitive end of Israel's exile, and the world's, comes with Jesus' ascension to his throne and the out-pouring of the Spirit. But the visible indication that Jesus has assumed his throne is the coming of the Spirit. So Pentecost vindicates Jesus' claim to be who he said he was. (The events of 70 A.D. do so as well.)
Pentecost isn't merely the beginning of the church, and it isn't merely the coming of the Spirit, although it is both of those. Both of those occur, however, because Jesus has ascended to the throne. And, as with Solomon, the Spirit does not come into the temple until the King has prepared the temple.
Daniel 7 gives us a peek into that day. Note that the coming of the Son of Man on the "clouds of heaven," is not about the Son of Man coming back to earth, but is about the Son of Man coming to the heavenly throne room to be received by God the father. So in the New Testament, when Jesus refers to "coming on the clouds of heaven," we need to read that, in light of Daniel 7, as a reference to his ascension, not as a reference to his return to earth in judgment.
Daniel's vision paints an amazing picture:
"And behold, with the clouds of heaven one like a Son of Man was coming, and he came up to the Ancient of Days and was presented before him. And to him was giving dominion, glory and a kingdom, that all the peoples, nations, and men of every language might serve him. His dominion is an everlasting dominion which will not pass away; and his kingdom is one which will not be destroyed" (Dn 7.13-14).
Jesus points only to his ascension in vindicating himself before the Sanhedrin (Mt 26.64, cf., Dn 7.13). His ascension to the throne is his sole defense before the Sanhedrin. For Jesus, everything rides on this one thing being true. So Pentecost is huge. Not just because of the pouring out of the Spirit, but because of what the pouring out of the Spirit means with respect to where Jesus is and how it vindicates who and what he claimed to be.
Jesus also points to his ascension -- the reestablishment of God's kingship over all of creation -- as prompting the spread of the Gospel throughout the earth (Mt 24.30-31). But this is also prophesied in Daniel as well. Jesus is given the kingdom "that all the peoples, nations, and languages might serve him" (Dn 7.14).
Finally, note the connection between speaking in tongues and the ascension: Daniel prophesies that the Son of Man is given dominion, glory, and a kingdom that "men of every language" might serve him (ibid.) According to Daniel, that's what occurs as a result of the ascension. And this prophecy was fulfilled immediately on the day of Pentecost when "men from every nation under heaven" hear the disciples speaking "in his own language" (Acts 2.5-6). So it is entirely appropriate that, in explaining the meaning of tongues (Acts 2.12), Peter preaches the ascension of Jesus Christ.