How can I make the most of this opportunity? Any suggestions for a 2nd/3rd year student interested in research?
Keep in mind, I'm not in math, so I'm not sure how much a lot of this will apply. 

I think if you're just starting out and trying to learn the ropes, I think an important first step is to get a feel for what research looks like in your field so that you're better positioned to conduct your own research independently in the future. If I were guiding a student in my field, I'd want them to understand: 1) how are papers structured in our field, and what type of information is generally included in most papers? 2) how does new research critically examine and contribute to the existing research? 3) how are empirical studies designed in terms of their methodologies? What considerations to investigators make in selecting participants, instruments/tasks, their methods of analysis?

Publishing is a rough process, but a reasonable goal once you have a strong understanding of what the norms are in your field. RE: writing to reiterate/further support existing findings...I'm not too sure. I think it depends; I feel like if you're further proving ideas that are fairly new/not quite established, it might work. However, if there's already a huge body of literature supporting said idea, then I don't think most journals would be interested in publishing that.