The circular version of the restricted three-body problem, along with the method of grid classification are used to determine the character of the trajectories of a test particle, which move in a binary exoplanetary system. The binary system can be either a parent star-exoplanet or an exoplanet-exoplanet or exomoon, while the test particle is considered to be an asteroid or comet, a space probe, or even a small exomoon in the case where the primary body is a star. By using modern two-dimensional color maps, we succeeded in classifying the starting conditions and distinguishing between bounded, escaping, and collision type of motion for the test particle. Furthermore, in the case of bounded regular motion, we further classify the starting conditions by considering their geometry (revolving around one or both main bodies) and orientation (prograde or retrograde, with respect to a rotating coordinate system of the primaries). For the initial setup of the test particle we consider two starting conditions: the launch from pericenter or apocenter. The final states are qualitatively visualized through two-dimensional basin diagrams. This approach allowed us to systematically investigate and extract dynamical information on the dependency of the test particle final state as a function of the particle's initial semi-major axis and eccentricity for a given primary and secondary mass ratio. Finally, we applied the restricted three-body model on several exoplanetary systems with observed mass-ratios and studied the dynamical behavior of a test-mass.