Great! The inverse Galois problem has always fascinated me.
A list of references is given in this answer. I particularly recommend Völklein's book, which is meant as an introduction. As he mentions, the basic techniques require only knowledge of introductory Galois theory, (finite) group theory, some algebra, and complex analysis since the theory begins by having groups realized via covering spaces of Riemann surfaces. The basic approach, dating back to Hilbert, who was the first to work seriously on the problem, is the irreducibility theorem, that ensures that groups realized over $\mathbb Q(x)$ are realized over $\mathbb Q$.
Deeper results require more background: An understanding of the theory of simple groups, for example, and techniques from algebraic geometry. There is the hope that the Classification of finite simple groups will allow an "inductive" solution of the inverse Galois problem, and this has greatly influenced modern research on the question. Some results, such as Shafarevich's realization of all solvable groups over $\mathbb Q$, are more ad hoc. Shafarevich's result is essentially number theoretic, for example.