Does it matter whether I learn OpenGL or Direct3D?
I don't think it matters much, which API you want to use when leaning to "program graphics". The more important things to learn are the typical concepts you use when working with 3D scenes. For example you want to learn how to write a simple (and later more sophisticated) Scene Graph. These concepts are much more important than the specific API method names you have to call. 

Compare it when learning how to write programs. If you are quite good at writing programs in e.g. Java, you won't have that much trouble learning C++, Objective-C, Swift, or some other object-oriented language. Because it's the concepts and the thinking which you learn.

The choice between OpenGL, Direct3D and Metal is primarily the choice, which operating system you target. Direct3D is primarily available on Windows, Metal on OS X and iOS, and OpenGL is supported on most systems including OS X, iOS, Windows and Linux.

The graphics card probably has little to no influence on this decision as I don't know a card that supports only one or two. If you have no dedicated graphics card, then rendering in real time will be a problem soon. Although a Intel HD Graphics and the AMD companion can already do quite much.

So, choose your platform.



Disclaimer: As of now, I did neither use Direct3D nor Metal.