In answer to your first question, yes you can CNC foam. People do it successfully with mills, routers, hot wire machines and probably several other machines that I haven't heard of. For your application I'm guessing a router is the best first step since hobby grade machines can easily handle foam, wood, and plastics.
Contrary to what you may have seen or heard neither a CNC router nor a CNC mill are able to make anything on their own. There are several other steps in the process that are always assumed by those in the CNC world and sometimes the line between the various steps will be very grey. I'll explain the process in reverse because you end with the machine doing the work but where you start can vary. You will likely be able to skip several of these steps but I'm including them for other beginners out there who plan to work with
Step 4: (this is where where you want to end up) You load a piece of stock into a "robotic"
machine and some type of cutting device removes material from the stock to bring it down to the desired shape. (router, mill, EDM, hot wire, etc…) FYI: You will need to learn a few tricks for fastening the stock material to the machine and how to tell the machine where you put it.
Step 3: You load a set of instructions into the controller that drives the machine so that it knows all of the movements that it will need to make to cut the part. FYI: you will need to learn how to operate the controller.
Step 2: You write the program that determines the movements that the machine will make. This programming language is called G-Code. Some machinists start here but the G-Code for engraving is usually written by another software package
Step 2 alternate: You use a software package that writes the G-Code for you. (MasterCAM, BobCad, FeatureCam, FlashCut, SurfCam, Gibbs, VisualMill and many others)
Step 1: You build a model of what you want to make using a CAD package. (AutoCAD, Rhino, BobCad…)
There are many specialized software packages on the market that can simplify and reduce the number of steps involved. If you are going to be building something intricate like a watch or a car engine then you might add a few steps to the process. A sign maker might use a single software product that handles all of the steps from layout of the sign on screen to controlling the cutter. You might want to consider SheetCam.
The machine is usually the simplest part of the process. You should check out several CAD/CAM packages and some PC based controller software. With BobCad and Mach3 (made by artsoft) you will have everything that you need to get started. Mach3 has a trial version that is free but is limited to small g-code files.
Side note: you will need to clarify the statement: Can I take pics from the internet, download them and have my machine make what I want it to make..?
You can take line art right of the internet, load it into a Cad package and 5 minutes later have your machine producing a part. But you have to realize the difference between 2D and 3D. This usually only works with 2D projects like engraving, signmaking, panels and sheets, and is easiest with line art but since CNC routers are designed for large flat areas it is very common. What you can't expect is to take a picture of a person and end up with a sculpture or take a picture of an airplane and have a toy come out of the machine. 2D picture = 2D result: easy. 3d stuff takes a bit more work.
With that said: You can accomplish everything you are looking to do with free software. It's just that without expensive software a business can't work efficiently enough to cover the overhead. Hobbyist's and small businesses can usually start out with the freeware and then work up from there. With better software more of your time is spent loading material into the machines and less is spent in Design/Drafting/CAM.
Checkout this link, halfway down the page it explains how make an engraving in 5 steps using free software. The author is using a mini-mill but the steps are the same for a router.
http://www.hossmachine.com/cnc_conversion.html#Testing