If civilization started in mesopotamia (correct me if I'm wrong), and then spread throughout the globe, when did we go from being one civilization in the middle east with presumably one language, to a species with many languages spread all over the world?

"Civilization" in this context means something like "civilizations with written texts". It definitely does not mean that all people spread from Mesopotamia.

Anyway, this is more of a topic for /r/linguistics. There is a hypothesis that all modern human language is descended from one ancestral one, which would be long before Mesopotamian civilization. It doesn't have much agreement in scholarly linguistic circles though--even if it's possible, there's no way to reconstruct it with any degree of accuracy. Instead different language families are reconstructed to a point in the past, beyond which establishing further connections gets shaky at best and is simply wild guesswork at worst.

So the short answer is that no, it doesn't seem so. The main issue is that all civilizations being descended from Mesopotamia is a misunderstanding.

After the last eruption of the Toba Super Volcano there was a period of time that the population of the human race dropped under 10,000 and potentially down to under 6,000 individuals by some estimates. This was a small population confined to a small geographical area. Depending on which theory of Linguistic evolution you subscribe to, this would be one of your better candidates for period of universal language.

If civilization started in mesopotamia (correct me if I'm wrong), and then spread throughout the globe,

The earliest "civilization" (I dislike that word, so scare quotes) was in Mesopotamia and it did eventually spread (including to Europe, which then spread it globally in the last 500 or so years, so in a sense all modern "civilization" can trace its roots to Mesopotamia), but there are at least six places around the world where "civilizations" developed independently of each other. There were people living in nearly every habitable spot on the globe when Mesopotamian "civilization" began, and there would have been several thousand unique languages. All people did originally come from Africa, though, so any primordial or ancestral language came from there.

This is also a good question for /r/AskAnthropology. Anthropology has our own branch of linguistics but it's also the discipline that focuses on the very long-term development of human biological and behavioural evolution, well beyond the scope of time that history can cover.

If you're at all interested in the development of human language, you should definitely check out the book 'Gesture and the Nature of Language'. It is a very detailed look at a relatively new and (as far as I'm concerned) vastly superior theory on language development and evolution in humans and our hominid ancestors.

Based on that book, I think its safe to say that some ideas are, or at least very close to being, universal across the entire human species. Since that's all language is, a medium to transmit an idea, this makes sense. Take pointing for instance; no matter where you go in the world, people will know what that gesture means regardless of language barrier. Another good example they use is making the shape of a pistol with your hand. Anywhere in the world that's had some kind of contact with firearms understands that gesture, and firearms are a pretty recent development within human history.

So I would say its entirely reasonable to say that at some point in the past there was a more or less universal human language, but the important distinction (made very clear in that book) is that it was a non-verbal language. Once humans developed the capacity for advanced spoken languages and stopped using primarily gesture based one's, that is when it is language would have split off into so many countless branches that gave us our modern languages.

All that being said then, by the time humans had developed "civilization" as we usually understand it, it is practically impossible for there to have been anything remotely resembling a universal language. Remember that by the time Mesopotamia arose as a society, there were people all over the Americas. There is absolutely no way ancient American aboriginals were speaking the same language as people in Mesopotamia. Theoretically, if we somehow found out they did, it would basically obliterate everything we think we know about language evolution.