A full 3D model is involved as you need some kind of 3D modeling software to create it.
With the 2D clipart you have to start with there are a few ways to go.
The easiest is 2.5D. That is the 2D art given some height. (Raised or recessed) You can mix and match different levels for a layered effect.
The Easel program is made for this. It now has an app that can trace images and you can them make adjustments to cut depth.
For more advanced 2.D creation and tracing I use Inkscape (a free program) to trace and layout the SVG file then import that into Easel.
Another approach is to use a V bit. This is a V shaped cutting but that will add that chamfered / beveled look to the carving. This is often done with the lettering on signs and create and very nice effect with minimal effort.
Because the effect is obtained by using a shaped bit you technically do not need any special software to do it. (you can still use ease, just lie to it about what bit you are using)
But you will need special software to preview what it will look like.
VCarve is one program that does this well. Though I believe there are some free stand alone programs out there for creating V-Bit gCode. (Who knows? Maybe someday Easel may support this as well)
Finally there is true 3D. This is a lot more complex as generally you will have to build the model from scratch using your original for reference. There are several programs out there. I have not had a opportunity to try them out.
Other commenters on this thread have done a good job of pointing out the most popular and available programs.
YouTube tutorials will be a great help here.
Once you have the 3D model you will need to convert it to gCode using a CAM program. Again there are several programs for this. I use VCarve (Free trial with select premade projects) .
MeshCAM looks like it may be a good inexpensive alternative (Free trial)
Finally sending the gCode to your mill. UGS(Free) is one of a few programs that you can do this with. It can be tricky to get working the first time, but is easy to use once you get things configured properly. There are several threads on the boards that outline this in detail.