How can I ease in to the research arena? I'm a first year graduate student and my age is 24. I got a degree in statistics, but it was not that interesting. I barely got into a below average grad school program in mathematics. I want to eventually publish some research in mathematical-physics in a few years but I doubt it'll be any good, because everything up to this point has shown me otherwise. I didn't "shock" on the SAT's. I've never been at the top of my class.

I want to look into special and general relativity, quantum information, gravity, and entropy and quantum mechanics. Relativity and quantum mechanics describe our world in very different terms. In quantum mechanics, events unfold against a fixed backdrop of spacetime — while in general relativity, spacetime itself is flexible. What would a quantum theory of curved space-time look like? These are the questions that interest me.

So here I am, in a mathematics grad program. Thinking about leaving and quitting math and science for good. But I think I want to give it one more go. Maybe research as a hobby. I do want to publish a paper in the next 3-4 years.

So, what's the best way for someone in my position to "ease in" to the research arena while maintaining a healthy work life balance?
Find an advisor and shadow them. 

With the help of an advisor find a problem that interests you. Plug away constantly and incessantly on that problem. Read a lot of related work. When you get some results decide, along with your advisor, what to do with them? Paper? Thesis? Repeat as necessary.

Once you have some success, replace/augment the advisor with other colleagues. Repeat. 

It is actually what about 90% (maybe more) of grad students do. There are a few that don't need the advisor, but I've not met any of them. The advisor is at least good as a sounding board.