What should I do or Where should I begin to become a good computer graphics engineer/programmer?

I will answer assuming you are talking about graphics programming. Graphics programming is about rendering a 3D scene with shading applied to it. There are two graphics APIs which can be used to write such programs, OpenGL and Direct3D. Pick any of these two APIs (however, underlying principles are same for both APIs), and start programming. You will find a lot of online tutorials that can guide you to write basic programs. Start with a "Hello Triangle" program and move on to render a cube. Then, there are shading techniques like gouraud shading, phong shading, etc. Try implementing those also. Once you are comfortable with the basic stuff, you are ready to move on to a little advanced techniques like shadow mapping, SSAO, etc. EDIT: For OpenGL API, OpenGL Programming Guide is a nice book. For in-depth understanding of real time rendering, Real Time Rendering, 3rd Edition is a great book.

I think it depends on what you want to do. Chirag's answer is a good one if you're interested in doing games and/or game engines or interactive systems. But there is little utility to OpenGL (and none at all to Direct3D) if you're interested in doing something like visual effects for film. In that case, you will want to learn a system like Renderman for imaging and/or one of the scripting systems built into interactive 3D systems (back in the day, the big one was MEL which was the modeling language built into the Alias system. Your mileage may vary). But in the end, the challenge is really not the programming languages or APIs. It's really just having (or developing) the ability to think in spatial terms, being able to visualize things in 3D space and implementing things based on that understanding. I spent almost 25 years doing computer graphics of all kinds, and that's what I really saw people struggle with. From my experience, about 30%-40% of the population will get there. The rest probably wont.

My advice, write a raytracer from scratch. It has such a shallow learning curve and you are constantly rewarded with better and better images, also debugging is very visual. You will learn a lot about vectors, matrices, lighting, physics, data structures/algorithms, optimisation techniques (BVH, kd trees, BSP, octrees, etc). You can also get into making it realtime, which is a great challenge and will lead you into GPU programming. A lot of what you learn will be useful for other fields, such as games or medical imaging. There is a good beginner article here: http://flipcode.com

No. We have artists to do our artwork :)  WHile it helps to be able to throw together "programmer art" for prototypes, its not essential.  Yo can either get proto art from your artists or scrounge it on the net. I do recommend at least one intro design course as it will teach you language that your artists use and make communication easier. Im not sure why you didn't grasp the graphics coruse concepts-- its all math.  Mostly Trig for 2D and Vector Algebra for 3D.  Id think as an engineering major you would have had vector algebra, yes?  if not, you need it.

Writing dots, picking a color, alpha color layer, copy-and-paste in Adobe Photoshop is enough. More important skill is making a good relationship to graphics artists. Don't focus to your portfolio. Even though you are writing an app which show a simple mesh object with texture and a directional light, it is more worthy if you made it from the scratch: your own model file format, your own simple model exporter for 3dsmax or maya and your own simple renderer (using Direct3D, OpenGL API directly, or your own software renderer). Regardless of which API you use, deep understanding of 3D graphics mathematics is more useful. It allows you understand harder technologies well. I have seen many programmers who cannot do anything other than copying and pasting shader code from books or googled results. If you have strong knowledge of 3D graphics theory, then you can create your own beautiful techniques.