If you want to be able to visualize things, learning how to draft three dimensional objects would help. This isn't just drawing or sketching, but learning how to draw realistic objects with standard orthographic procedure (you could learn it in an afternoon). This image right here should give you the general idea. There's more to it than this, but I'm not going to pretend that it's hard or extremely complicated.
As you can see, there is a standard way of breaking down a three dimensional object into three one dimensional drawings. It has its own format (hidden lines are represented by dashed lines, etc). This process, more than anything, has helped me visualize three dimensional objects. You should be able to find the topics related to orthographics, third angle, first angle, isometric design, (first, second and third) perspective, oblique projection, that kind of thing. Google should be able to find all these things for you.
An aside : Learn with a pencil, ruler, and grid paper first. There is no reason to jump into using CAD software if you can't even command a pencil to reveal what should be pictured in your mind's eye. I have spent enough time drafting by hand (which wasn't even that much time, maybe 10 hours total) that I have no problems visualizing between third angle (the three one dimensional pictures) and what the object looks like in 3D.
If you want to think more logically, take a foundations of math course where you learn how to prove theorems. Set theory and statistics are also good subjects to study and even an intermediate knowledge of them will grant you new perspective that most math-illiterate people would never figure out in a hundred years. There's a reason why I will never buy a lottery ticket, sign a mortgage or buy a brand new car and it's not because I'm poor.
A proper debate class (or club or whatever) is also smart. Knowing how to prove things on paper is great for justifying your teacher's salary or writing spreadsheets, but learning how to set up an argument and properly defend it takes a different kind of reasoning. This isn't the same as persuasion; that is a different subject entirely.
Outside of that, find a fun game with a puzzle element.
edit: Almost forgot the most obvious answer to the second part of your question.