Well, to start with, there's Agent Mulder and Agent Scully. The Believer and the Skeptic. Now, you don't want to take those descriptions too far and make their entire personalities based on that - it would be silly, for example, for the Agent Scully to still deny some weird things happen after seven seasons of investigating aliens, monsters, et cetera.
There is the choice of who your Big Bad Ancient Conspiracy Whatever might be. For The X-Files, it evolved over the years - first, it was the mysterious Cigarette Smoking Man. Then there was the Syndicate, then the Rebel Aliens and...well, it got complicated, so just trust us on this. For your show, however, it doesn't have to have anything to do with aliens at all. Make it something supernatural or just plain ordinary (but not too ordinary).
See also Write a Conspiracy Theory for more tips.
Here's something you can't watch out for: Executive Meddling. Any time you start a series with a potentially huge and long-term arc, particularly one steeped in mystery or character development, you run the risk of getting cut, getting messed with, having your most inspired ideas shot down in flames...
About all you can do about this is write as good as you can in the first place, and make backup plans. If worst comes to worst, you can reveal a lot of this through Word of God in places other than the show itself, especially given the options provided by the Internet.
Now, about your plotting: Try to decide whether you're a giant Myth Arc series or a one-shot Monster of the Week series. While it's possible to have both, it's also likely to result in a muddle. (Still, you might play those one-shot monsters as a Chekhov's Gun: Wild Mass Guessing has concluded that should those aliens in The X-Files ever attempt to take over the world, they'd be soundly whupped by all the monsters that are already there.)
If you're going for a Myth Arc, it's a good idea to try and plot out some idea for how you want the arc to proceed over the course of the series — The Chris Carter Effect is named after the creator The X-Files for good reason, since it can quickly become apparent if you're just making stuff up as you go along or if you're just stringing a plot out without ever intending to (or knowing how to) resolve it. Obviously the realities of media production and the simple fact of changing your mind can mean that you might not be able to stick to this plan without making any kind of amendments at all, but it's a good idea to at least know generally where you want to end up in advance.
The Agent Mulder turns out to be a traitor halfway through the first season. His job was to be a paranoid nut, and make The Conspiracy look ridiculous. This happens shortly after the team appears to succeed in exposing The Conspiracy. In truth, a few core members of the group had Agent Mulder expose their rivals, and continued their work in secret. After all, there COULDN'T be two alien-related conspiracies on Earth.
Suggested Themes and Aesops
Trust No One is always a good one. In fact, much of The X Files was invested in horror tropes of various sorts, which could be Paranoia Fuel designed to teach you to never be curious, never leave the safety of your hometown, never leave the safety of your house, get out of your house, it's not safe....
Set Designer / Location Scout
The X Files ranged through a wide variety of indoor and outdoor sets, all set on Earth (well, there was that dream about Mulder on a spaceship...).
Guns and Gunplay Tropes, of course.
The X-Files had this neat gender-switch going on. See, generally speaking, when you have an Odd Couple and it's split between Facts and Beliefs, the man takes up the banner of Facts and the woman takes up the banner of Beliefs. Not so here (except when Mulder and Scully argued over religious beliefs).
Rounding out the group is Da Chief, Skinner, who eventually rounded out as a character who got emotionally invested in the success of his team and wasn't above a little fisticuffs when occasion called for it.
A bit of gunplay and plenty of chase scenes.