In this article we present, as a case study, results of undergraduate research involving binomial coefficients modulo a prime p. We will discuss how undergraduates were involved in the project, even with a minimal mathematical background beforehand. There are two main avenues of exploration described to discover these binomial identities. The first uses a p-adic approach. The second, and more modern, utilizes the self-similarity, i.e., fractal nature, of Pascal's triangle modulo a prime. We will also discuss strategies for involving mathematics students in undergraduate research and provide some ideas which can be incorporated in several of the standard courses in the undergraduate mathematics curriculum. In particular, the identities and techniques discussed can serve as a bridge between classroom activities and future undergraduate research projects.