Will preparing for the mathematics GRE help me towards my goal of doing research in applied mathematics?
The mathematics subject GRE covers roughly the first half of the content that one would learn in a good undergraduate math program.  Much of it (calculus, linear algebra, differential equations) is material that an engineering student would likely be familiar with already.

I think that the only good reason to study for the GRE is if you need to take the GRE, i.e. if you want to apply to a school that requires it.  In that case, it is probably best to spend a limited amount of time studying in the few months or weeks before you take it.  Otherwise, I think your time would be better spent on more advanced topics.