I won't be applying for master's programs until next year. Should I start studying for the GRE now? I'm an undergraduate in mathematics, and won't be finishing my Bachelor's degree until next Spring or next Fall (2016). I plan on applying for a master's program before possibly moving onto a PhD (if I like the research) after I get my bachelor's.

Should I start studying GRE-type material now, even if just in my free time, to hopefully have a bit more of an intimate knowledge of the kinds of questions it will ask, as well as really solidify my problem solving in those areas? Or is that overkill?
First, the GRE, especially the subject test in math, is a very highly stylized version of a very particular slice of a fairly low-level piece of school-mathematics... Whatever one thinks about it, it is widely used, and its idiosyncrasies and limitations are often over-looked for the sake of the convenience of having a single number to (supposedly usefully) compare people from very different backgrounds. Thus, you don't want to show up and be surprised by what it is, for sure! As in comments, do some practice exams to see what the questions are...

Also, do think in terms of the possibilities for "gaming" a multiple-choice exam. The most successful of your competitors will have some success with that aspect, apart from knowing or not-knowing mathematics.

But do not let the GRE too-much affect studying actual mathematics. For example, things that don't admit "testing" by multiple-choice questions can't make it onto the GRE, yet arguably are far more important in the larger enterprise than things that can.

The possibility of gaming the exam also means that "problem solving" is not really what it's about. It's about doing enough diagnostic stuff to eliminate wrong choices, and then whatever's left has to be right. This is methodologically wildly different from the practice of mathematics.