I am not very much into the Harry Potter books but have watched every part of the movies like a hundred times and couldn't find how Snape ended up becoming the servant of the Dark Lord for the very first time.
We're not told exactly how, but it's likely that he was recruited while still at Hogwarts. Sirius tells us in Goblet of Fire that Snape had a long-time fascination with the Dark Arts, and that he hung around a bunch of future Death Eaters:
Snape's always been fascinated by the Dark Arts, he was famous for it at school. Slimy, oily, greasy-haired kid, he was," Sirius added, and Harry and Ron grinned at each other. "Snape knew more curses when he arrived at school than half the kids in seventh year, and he was part of a gang of Slytherins who nearly all turned out to be Death Eaters."
Goblet of Fire Chapter 27: "Padfoot Returns"
This was a point of contention between him and Lily, in fact; at one point she accuses him of being eager to join up with Voldemort:
You and your precious little Death Eater friends – you see, you don't even deny it! You don't even deny that's what you're all aiming to be! You can't wait to join You-Know-Who, can you?"
Deathly Hallows Chapter 33: "The Prince's Tale"
Whether he joined while still a student, or shortly after graduating, is unknown.