Hey,
A good entry into very basic CAD can be found in Tinkercad (tinkercad.com). That's drag and drop CAD with support for 3D printing and not much else...if you want to start machining stuff that's probably not powerful enough. If you're a student, that's awesome because companies love to give away their multi-thousand dollar CAD software--Autodesk Inventor or AutoCAD is completely free, just enter your email, and SolidWorks also gives out their products but the process is harder. I have experience in Inventor and SolidWorks, both are good and whatever you can get your hands on is what I'd go for. Inventor is probably easier to access, and also has really accessible CAM (Computer Aided Machining) as well-which is the intermediate step between CAD and machining.
The process goes CAD-->CAM-->G-Code (generated by CAM) --> machine.
With laser cutting, every laser cutter is different but what pretty much all laser cutters intake is a DXF file. Check with whoever supervises/runs these tools to understand the procedures.
CNC machining is very complicated and has lots of aspects. A good website to explore is cnccookbook.com, which has a ton of information about almost every part of machining.