To a broad overview, it would be pretty close to what the XCOM game was like, except that first there would also be an X-Files division telling the X-Com jarheads where to point their guns and second there would be multiple different X-Com organizations, at minimum one each for America, Russia, and China, and they would work together to greater or lesser extent based on the situation. Which also means there would likely be infighting and paranoia. Is the Russian X-Com breaking ties because they've been infiltrated by aliens, or are they just being dicks?
The aliens have an overwhelming technology advantage but their numbers are incredibly limited, making a straight invasion impossible. Instead, their plan is to use mind control to take over world leadership invisibly and dupe humans into being their occupying force. So the X-COM guys work in small, 5-10 man units and have total numbers somewhere in the neighborhood of a few thousand, most of whom are base defense personnel. This makes it relatively easy to weed out infiltrators as opposed to a millions-strong united Earth army, and they only need to be big enough to kill sectoids mind-controlling VIPs, punching through that VIP's private security (possibly including government soldiers/police who either work for a non-compliant government or are willing to break ranks on their bosses' command) and any aliens sent in to help. Depending on the VIP and how cooperative the local government is, you might need anywhere from a single 6-man X-COM squad on hand to take over if local government forces break down to a 30-man platoon launching an assault against a hostile government facility, but either way you don't need infantry divisions and you don't benefit from having a much larger army that's far more vulnerable to infiltration.
Since humans do have a decisive numerical advantage, however, it is the aliens, not the humans, who will need to pick and choose which operations they want to reinforce, while X-COM can send a squad to thwart every single abduction attempted. Also, it is unclear why they attempt abductions, but I am assuming the aliens need abducted humans as part of their infiltration plan.
Now, terror missions are where things get weird. In X-COM, a terror mission is a huge abstraction. It's played up like it's basically an invasion of a city, but the total number of aliens is like fifteen and the total number of doods sent to stop them is like five. And the city being invaded is protected by an army, which can crush the invading force under their numbers (unless they can't, in which case the aliens don't need to be all sneaky and subtle all, they can just launch a global invasion). So the realistic version of this would be that the aliens are presumably dispatching several thousand aliens to try and cause havoc before government forces can show up, and if X-COM helps it's mostly in an advisory position, because the member nations can use their own damn armies to defend their own damn cities. They probably have closer bases anyway. There is a serious question, however, of why the aliens even want to conduct terror missions. In the game they arbitrarily cause council members to lose faith in X-COM and withdraw, but that only makes sense if they expected X-COM to defend them instead of their military, rather than in addition to their military. If you've just had one of your cities ripped apart, why would you then choose to back out of a defense pact? Terror missions might make sense as an attempt to cause political instability in order to give alien-controlled VIPs the opportunity to seize the public narrative by being strongly anti-alien and criticizing the government for failing to stop the attacks. That's a risky ploy, since in X-COM 2 the aliens' plan is to rule openly. Terror missions just don't meaningfully advance the aliens' objectives in the first place.
Since we don't have any means to read an alien species' mind, the plot of mind-fucking aliens to find secret base locations isn't going to happen, but the X-Files division might be able to locate them on their own. And an alternative route to victory would be to just wear the aliens down with attrition, thwarting their attacks until they no longer have any sectoids left to mind control world leaders. Really, when it comes to available resources and effective strategies, the aliens and X-Com are almost exactly backwards in the actual X-COM game. X-Com has effectively unlimited numbers and can deploy to far more hot zones than the aliens. The aliens are much more powerful individually but must pick and choose their battles and too many botched missions will cause attrition that they can't recover from (although the aliens probably won't suffer the "one failed mission will doom us all" sort of attrition problem that X-Com faced in the games).