I think this is a really great goal. I encourage you to pursue this. Just understand that it’s going to take a while — probably a couple of years, at least — so you will want to work up to it, and do some easier games first.
Yes, you’ll want to set up a version control system. I recommend Subversion (aka svn), and as for where to host it, I recommend Deveo. Deveo has a nice introduction to svn. Be sure to read this page of the Unity manual.
In that case you should abandon this idea. Maybe you guys find some easier hobby.
On the other hand, if you have what it takes — which is mainly patience, determination, and persistence — then you can absolutely do this. The professional FPS games you’re probably used to are made by teams of 30-40 people, working full time for several years, with a budget of several million dollars. Five youths in their spare time are not likely to replicate all of that in a few months without a willingness to even learn the necessary skills.
BUT, if you set your expectations appropriately, spend some time building the skills you need, and learn how to leverage assets available in the Unity community, in a few years you could make your own FPS that is fun to play, and unique in the world — nobody else will have an FPS exactly like yours. That’s pretty cool. And you’ll have built wizard-like skills that will enable you to do lots of other things that are even cooler.
Unity is absolutely the right choice for your game dev environment. For modeling, especially on a tight budget, you’ll probably use Blender. Learn a Photoshop-like app (I actually use Affinity Photo) for image editing. Finally, for code editing, I strongly recommend buying Script Inspector 3, as it makes coding in Unity so much more pleasant.
Good luck, settle in for the long haul, and enjoy the journey! And let us know how it goes.