Why Pentecost? Why did the Spirit come on the church on that day in particular?
It's a good question. Jesus ascended on a Thursday; the Spirit did not come for another 10 days. Why the wait? Jesus only said that they'd have to wait for "a few days", but 10 is longer than it might have been.
Perhaps it was just expedient. There would be multi-national crowds in Jerusalem over the Pentecost festival, making it a perfect time for them to hear the wonders of God in their own tongues.