I managed to run a pretty successful X-Com game awhile back (before even the X-Com PC reboot).
I used Gurps Black Ops and Gurps Conspiracy X 3e books as my base campaign, especially for the look, feel and tenor of the campaign.
Blacks Ops introduces an "academy" where they train the soldiers, so I used their "graduation" scenario as an "alien free", first adventure.
The second adventure where I introduced the aliens, it was more like a horror campaign, with them being shot at from the dark, and only seeing scuttling, strange shapes.
Go checkout Black Ops:
http://www.sjgames.com/gurps/books/blackops/
It has a lot of what you want: "watch out for the insidious Greys, as they kidnap human women and steer the destiny of entire nations!"
It is 3e, so you have to convert it. But I think it is a great base to build on.