What was the first language actually spoken in the Neolithic era? Into what languages did it eventually evolve?

there is no definite answer to your question.

The Neolithic era in generally considered to precede the Agricultural Revolution by a few thousand years and, according to archeological studies, is dated from approximately the year 10 to 12 thousand BP (before present).

There is no doubt that humans communicated at that time with a spoken language. We are referring to people capable of drawing such wonders as the Lascaux and Chauvé caves. They certainly had words to express those feelings. Unfortunately in the Neolithic script had not been invented yet and, without a written or scribed testimony, there is no way we can access their languages. Written words firstly appeared with the cuneiform clay tablet in the third millenary (in Mesopotamia) well after the Agricultural Revolution and the urbanization process.

At any rate there are theories speculating the existence of an original "mother" language spoken by Neanderthals 70,000 years ago. There are tons of articles on the subject on the net. My opinion is that it is difficult to speak of a language without a nation identifying with a common and structured state.

The cave-men's means of communication had to be as different as there were communities. One needs a unifying organisation to build a common language and that did not come until after the Neolithic, the "metal" eras and the Agricultural Revolution.