Welcome to the wonderful world of graphics programming! I'll try to answer some of those questions for you.

1. Learn C++:

Any language which has access to a well-documented graphics library will do, so you don't necessarily have to go for C++. Honestly, if this is your first experience with programming I'd actually recommend going for something like C#. There's plenty of awesome libraries or wrappers available to do graphics related programming and it'll allow you to get up and running more quickly. It's going to be a while before you can get any real benefit out of working in C++ anyway, so it might be a smaller hurdle to get over by sticking with a more "approachable" language. I know a lot of people have different opinions on this though, so I'll let others add their opinions to this as well.

2. Brush up on Linear Algebra and more specifically, have a solid understanding of the use and applications of Vectors and Matrices?

Absolutely! Understanding linear algebra and trigonometry is going to help you out a lot if you want to make any progress. Once you advance a bit further it can become useful to have a good grasp of some elementary concepts in calculus, like differentials and integrals. Once you're at the point where you need those you should have an understanding of the following mathematical concepts you're going to need to advance. It's best not to worry about very complex things for now, just basic linear algebra can already get you far.

3. Choose between DirectX or OpenGL Personally I would probably pick OpenGL as it's crossplatform, so what comes next is understanding the entire openGL pipeline, its libraries, etc? glsl, shaders?

The fact that the OpenGL standard is supported on multiple platforms does not mean that it will work out of the box across multiple platforms or even across multiple graphics hardware vendors. OpenGL is notorious for being very inconsistent in these cases. If you're starting out I wouldn't focus on getting things to work on multiple platforms just yet. Start off with one and see how far you get. Worry about the mess that is cross-platform graphics development later.

If you want to focus on actually implementing graphical techniques (i.e. you actually want to write code and shaders to get thing to show up on screen asap) I'd actually recommend going with something a little higher level than plain DirectX or OpenGL. I'm a bit out of the loop on publicly available graphics engines, but there's plenty of them out there. I know libraries like Ogre used to be very popular, but I haven't had a look at any of those libraries in years. Even working in Unreal and Unity for experimenting with shaders and such can be a great introduction without too much stuff getting in your way.

I'd recommend looking at OpenGL and DirectX once you understand the needs and requirements of graphical applications a bit better, or if you're just really passionate about having a look at more architectural stuff.

The most important thing is to experiment a lot and to not worry about not getting things right immediately. Don't be intimidated by the complex looking math and such. Find sample implementations for things, play around with them, try to understand them and try to connect them mentally to their mathematical description.

Good luck!

Are you trying to get into the demo scene or real-time video game graphics or offline rendering?

Here is something I've been slowly digesting, its used in the demo scene. http://9bitscience.blogspot.com/2013/07/raymarching-distance-fields_14.html

When looking around online, some people have said that it's better to start out with C as C is a lot smaller in comparison to C++ and is stricter, which makes C a little easier to learn. But, on the other hand, there's also the opinion that starting with C sometimes makes people fall into bad habbits when they do try to write code in C++ as they often fall back to writing in a more C style code. is this true?

Being C++ an extension of C, at the core there's really not much difference between both of them... C++ just adds new stuff to the language, abstracts some others and fixes a few little things here and there, to the point that you could almost compile C code on a C++ compiler without problems... so, by learning C you're already learning the basics of C++...

Also C makes you think more at the metal level, you could easily figure out the assembly produced by your code in C... and that's good... if you then move to C++ and see how it abstracted some things, you can figure out what's going on on those abstractions (nothing is pure magic), and you can decide if it's good for what you're doing or not...

But don't give it too much thought, as long as you understand what's going on with the code you write no matter what you learn first or after... the important thing is to take the next step in your career ;)