I agree that you should learn the underlying concepts, I would check out the nvidia opengl sdk, it has a wealth of knowledge (but you must understand that those are quicky demos and if you wanted a full engine you wouldn't mimic some of the things they do).
I also highly suggest checking out Humus' Framework3, It is a great reference for looking at the ins and outs of the various apis.
You can start in any or all of the following,
The Cg Tutorial by nvidia,
The Orange Book (GLSL), (get the latest addition)
The DirectX SDK (HLSL)
Shader model 3 (dx9) is the most common shader model you will find in hardware today.
If i recall the orange book is great for those leaving the fixed function world and entering the programmable pipeline world
Ok first step is as you are, learn to code, C++ is a good language for it, but as jump starting for beginners oriented to game development it wouldn't be my first choice.
I would advice you to begin with a more game oriented framework such as Flash ActionScript, Unity or XNA, once you have the basics of object oriented game programming you can migrate to other languages with more ease.
I agree, and I cannot stress enough how important it is to do this. These environments are controlled environments, ie. it is more difficult to cause a problem.
Unity, XNA, all provide you with a foundation for making games, a community to ask questions, and tutorials galore at your disposal. This may not be the exact technology you want to use with your game (i actually think they could be very easily you just dont know that yet, especially unity), but it is a great foundation to help you learn the basics of coding in games.
Couple this with learning C++ on the side, and you would be very surprised how fast the 2 will meet together. You will learn what classes make up a framework, what parts of a framework you REALLY like, (ie automatic asset loading from a folder on your computer) and see how you would use them in making a game. I cannot say how important this kind of knowledge is when coding a game engine of any kind.