I'm still making the components list that I need. I'm a student and as my final project I had the idea of making this machine. This machine's objective is to create a Fablab in my school.
The money isn't really a problem, since I already presented this project to my school and they accepted to finace it.
One of my teachers is following this project and helping me developing it. He talked about the motor strenght and precision problems too.
The idea is to use a threaded rod instead of a belt in order to achieve the precision and strenght needed. As for the motors, I think that all I need to do is to buy one that has better strenght.
Do you think it'd be possible to succed following my way of thinking?

Thanks for answering, cheers!

For the machine to be able to CNC mill, you are going to need the machine to be extremely heavy/sturdy/rigid. Both the structure and the gantry.

With that requirement, you end up with a machine that has a *lot* of weight ( which severely reduces acceleration and therefore speed ), high-torque ( and therefore slow ) motors, and leadscrews ( also slow ).

This means that if the machine is good enough to do CNC milling, it's going to be insanely too slow to do laser cutting and 3D printing at any reasonable speed …

It's a problem you can only solve by throwing insane amounts of money at it ( extremely oversized motors on a heavy steel cnc gantry, solidly bolted into the ground, could get you somewhere )

Speed, Cheap, Quality. You can only get two of those, not three.
Speed and quality won't be cheap. Speed and cheap won't give you quality ( ie no CNC milling ). Cheap and quality won't give you speed ( ie no 3D printing/laser cutting ).