Material choice really depends on the application.
Generally speaking, plywood is an excellent choice for structural applications, as it is strong, relatively dimensionally stable (it doesn’t swell like solid wood), and can get wet without getting destroyed. However, you can have problems with tearout much like solid wood, which can ruin your finish. Also, because it is made up of layers of wood, it does not sculpt very well. Finally, plywood has a tendency to warp over time, especially in humid environments.
For my own uses, I primarily use plywood.
MDF, on the other hand, takes detail really well. You can pretty much sculpt using it, especially if you glue multiple layers together. It is even more dimensionally stable than plywood. It does not have the same tendency to warp like plywood does. It does get obliterated by moisture, as others have mentioned. In my experience, it will actually start to fall apart if it gets too wet. It is also VERY heavy. Dust is an issue with MDF as well. A good dust collector should mitigate that, however.
Particle board has similar properties to MDF, except that it doesn’t have the same level of surface finish. Generally speaking, it’s more a substrate for laminated boards than a finished material.
Both MDF and particle board are great candidates for slab doors and other parts that are self-supported and need to not warp.
I have not machined solid wood on my router yet, but I don’t see any reason why you wouldn’t be able to. Unlike plywood, MDF, and particle board, you don’t need to cover the edge grain with banding or nosing. But you often need to glue multiple boards together to get a wide enough panel for most CNC routing. Also, it is the least dimensionally stable material to work with and will warp easily if you’re not careful.
There are also various types of plastics that are good for machining. Generally speaking, they are going to be more expensive than wood products. They do sculpt very well, so long as you have your machining parameters set right and the correct tooling. A lot of plastics are thermally sensitive and if you cut too slowly it will melt and give you a bad surface finish. I have less experience in this area, but I estimate that I will be working a lot more with different types of plastics in the future with my Maslow
Maybe I should put some of these observations in the wiki?