Justin - following up on your comment above. I agree with what Gilrock was saying.
Mach3 is just the software that controls the CNC router. It's rather an older (and somewhat more hobbiest oriented) system but works just fine. It takes G-code and converts it into movements on the CNC. To toolpath you will need to bring your DXF into another software package (like Vcarve, Aspire or similar) and provide information about the material and thickness. Every CNC machine has its own capabilities which are a function of spindle power, size, presence or absence of an automatic tool changer, etc. I'm sure if you are taking the training they will cover this although 90 min seems a bit quick. You need a postprocessor for Vcarve/Aspire that is specific to your machine and some understanding of tooling and holddown issues. You can easily crash the machine, break bits, or screw up your workpiece if you don't understand certain aspects of toolpathing. So there is a bit to it but it's certainly fun work. Solidworks certainly also has add on manufacturing modules that will allow toolpathing as well directly from your model but I have no experience doing that with solidworks (which seems more focused on CNC milling, etc).
Most of the work in this case will be in the planning and setup stages. Cutting out something like this on a decent CNC will take you < 10 min. You may want to consider how you are hinging things. If you are going to use Euro-style hinges (or anything really) you may want to have the CNC mortise out the cups and drill holes, etc. Or you can go easy and just do all that manually afterward.