As you all know, we follow a different canon, which is why we're an AU, an alternate universe. Anything you know can be changed here, so please come here to learn of the differences.
Firstly, the first X-Men were indeed Beast, Charles Xavier, Magneto and Mystique. No one else.
They didn't fight the Hellfire Club, they faced instead the prejudice against mutants, and that's immediately where the conflict between Charles and Erik began. After humanity tried to strike the X-Men, the team attempted to defend themselves, only for Magneto to finally decide to end humanity, although Charles attempted to stop him.
The events at the end of the movie X-Men: First Class are very similar, after the X-Men are cornered into an island, Moira shoots at Magneto to attempt to stop him from nuking the humans attacking them, and Magneto then paralises Charles with a bullet being deflected to his back.
Magneto then leaves with Mystique, beginning the Brotherhood there, while Beast and Moira remain with Charles.
From there on, Charles formed the X-Men, recruiting Jean Grey, Storm, Cyclops and Beast, making them the first X-Men squad. With the coming of the years, more X-Men joined, and eventually both the X-Men and The Brotherhood went after Wolverine, for different reasons.
The Brotherhood wanted to help Sabretooth destroy Wolverine, while the X-Men wished to recruit them. The X-Men were successful and they became the team you know and love.
In our universe, groups like the Hellfire Club never surfaced, and the X-Men never fought them. The Sentinels have yet to be created, and Stryker was never killed. Apocalypse never appeared, and neither did Mr. Sinister. The only famous arc that happened in the past was the Phoenix Arc, that took the life of Jean Grey and Charles Xavier.
Before all that, the wars were between mutants and humans, and between the X-Men & the Brotherhood.
Now, those evils begin to surface, beginning with the newest group, an original one, called The Shadow Collective. The rest will surface soon enough.