I don't think becoming an expert in neuroscience in your spare time is a realistic goal. On the other hand, learning enough to understand basic neuroscience research is feasible. Neuroanatomy would be a good place to start. Know all the brain structures, their function, and their interactions with other structures. Know what a neuron is composed of, learn the types of neurons, and the supporting cells/structures of the nervous system.
Chemistry is extremely important, including physical chemistry, organic chemistry, and biochemistry. From there, physics is useful (kinematics, electricity and magnetism). In fact certain brain processes can be modeled as electronic circuits. Math (Calculus, Differential Equations, and Linear Algebra) comes into play at times.
I listed these subjects out of order. For context, during my degree program, we spent the first 2 full years studying chemistry, biology, physics, and math exclusively before we even learned a single thing about the brain.
Neuroscience is a very multidisciplinary field, and the wide variety of approaches to neuroscience research reflects that.
Also be wary of "neuroscience" papers that are really just psychology papers. They tend to be light on math/chemistry, and heavy on behaviorial data.
I got interested in the field in high school after reading some books by Oliver Sacks and Marvin Minsky, two prominent former neuroscience and cognitive science researchers. You might like their books too, they tend to be more accessible to the layman.
Since you're a student, why don't you look into minoring in neuroscience?
Understand what the earliest principles were and how they evolved into today’s principles. Find books written by the early pioneers or those who can introduce you to them.
Enjoy the journey but if you’re looking for an easy way, there’s none.
Reading 1-3 hours a day will get you really far in five years, further than those with degrees. Couple your reading with 10-30 minutes of writing and reflection and you will be a master by no time. Share your favorite findings at the end of the week in a podcast and newsletter, and buddy, you will be having lunch with Andrew Huberman in a few years.