@Brandon9000,
With only a few hundred stars to search, it wouldn't surprise me if we didn't find any life forms at all, and it would astound me if we were so lucky as to find even one civilization of intelligent creatures.
How many planets would be circling a few hundred suns? I think 10,000 would be too high a number. We could hit the mother-lode and find the home planet of a galactic empire, but the odds against that would be…
astronomical.
Space is unimaginably vast (you watch Cosmos) and a period of decades is a less than the blink of an eye. But, we could receive convincing evidence upon waking tomorrow. If it’s ever going to happen it could just as easily be tomorrow as a hundred years from now.
I'm not sure that the mere certain knowledge of the existence of life beyond earth would have the sort of profound effect on us that many people seem to believe. I suspect that most already take it as a given that there is life out there, and learning that Planet 9 has single cell organisms living in pools of sulfuric acid isn't going to be all that impressive to the majority of humans.
It could stir up interest in space exploration that has been lying dormant for some time, but whether anything came of that would be dependent upon the conditions on earth at the time of the discovery. A world preoccupied by a major war, or a global economic crash probably wouldn't have much time or energy to think about going into outer space.
I can imagine theologians giving it a whole lot of thought in terms of how it conforms with their religious beliefs, but I don't see the average Catholic or Muslim having a crisis or renewal of faith.
If the evidence was of an advanced civilization, it would have more of an impact but most likely in terms of the anticipation of contact. The classic idea of cults forming to worship the aliens probably would bear out to a minor extent, and survivalist products would enjoy a boom in sales, as would anti-depressants, while cottage industries based on what the aliens may look or be like would thrive.
To the extent there was any noticeable impact of the way we think, my guess is it would involve a pervasive existential dread rather than some material alteration of the way we think about "our place in the universe." Either way it wouldn't usher in a new age on earth.
It would certainly cause an initial huge splash, but in terms of extended impact, not much.
If they come to visit, that's a different story.