Did the gospels really copy from each other? In school, I learned that the different gospels did copy a lot from each other. This was the explanation for why there are so many overlapping events between the gospels.

Is there substantial evidence to support this besides the bible?
Yes, although not quite the way you put it.

It is agreed by pretty much all Bible scholars that there is some degree of interdependence between the gospels. This is based on the fact that many of the stories in the gospels are told in a very similar way. Exactly what depended on what is the subject of much debate, and more information can be found on Wikipedia and Catholic Answers. This belief is very old, with Augustine laying down one of the most widely believed dependencies. 

Some of the beliefs of dependency include:


Augustinian view: Matthew written first; with Mark using Matthew, and Luke using Matthew and Mark
Q source Matthew and Mark used a now lost source called "Q". Luke used both Matthew and Mark
Markan priority Mark was written first, with Matthew using both Mark and an unknown source
Four-source Mark and three unknown sources were written first, which the other gospels drew on.


It's worth pointing out that the interdependency has never been seen as in any way invalidating the gospels as sources. Any thorough modern biographer would also make use of other documents about their subject, and the parallel passages can be seen as evidence of the thoroughness and accuracy of the gospel writers. it's worth noting that all the gospels also contain some material found in no other gospel. Clearly none of the gospels was composed simply by copying other gospels. The wording differences are great enough to rule that out. Nor does it rule out the idea that the writers were eyewitnesses of at least some of the events themselves.

In answer to your final point, the only evidence for this is from analysis of the gospels themselves, though that includes the dating of manuscripts of the gospels and fragments of the gospels. None of the hypothesised sources such as Q has been definitively identified.