Diving straight into "research" versus studying broadly as an undergrad (soft question)
[This is really a comment, but too long to fit in a comment field.] 

You seem to be exceptionally well prepared compared to most students at the end of your freshman year.  And I'm not sure what you mean by "research" or "publication".  But, by the common meaning of the words in theoretical CS, I would say that the goal of a PhD is to learn to do independent research leading to publication, and that, as a rule, achieving that goal requires a lot of hard work, time, and usually help from an advisor, spread over more than four years of graduate study.  This is why a typical REU (research experience for undergrads) does not have this goal, but instead will allow you to work as a team member in a larger group, or perhaps with an advisor on a smaller project, with the more limited goal of gaining some initial exposure to the process.  (Even MS students rarely do original, published research.)  Can you find any summer REU opportunities to apply for?  They could be at another school.

I'm not sure what to say about "depth" vs "breadth".  I would guess that, at your stage, your fastest way to develop is to complete your undergraduate TCS courses, and take some graduate-level TCS courses if you can.  You may be able to work through some of the undergraduate material on your own in the summer (maybe that's your "studying broadly").  Meanwhile, if there is some degree of TCS community in your school, or nearby, you could try to connect with it.  Find students (including grad students) and a few faculty to cultivate relationships with, and find out what kind of research projects are going on nearby.  A big part of learning to do research is becoming part of the TCS community, learning its norms, and what is currently of interest.  Over the years I've known a small number of students who had an idea they wanted to pursue, and tried to start research on their own, and while I'm sure some have enjoyed the experience, the usual outcome (by an academic / professional standard) is to fall short, for all kinds of reasons.