This paper presents a numerical investigation of the dynamics of pinch-off in liquid drops and jets during injection of a liquid through an orifice into another fluid. The current study is carried out by solving axisymmetric Navier-Stokes equations and the interface is captured using a coupled level-set and volume-of-fluid approach. The delicate interplay of inertia and viscous effects plays a crucial role in deciding the dynamics of the formation as well as breakup of liquid drops and jets. In the dripping regime, the growth and breakup rate of a drop are studied and quantified by corroborating with theoretical predictions. During the growth stage of the drops, a self-similar behavior of the drop profile is identified over a relatively short duration of time. The viscosity of the drop liquid shows substantial influence on the thinning behavior of a liquid neck and a transition is observed from an inertia dominated regime to an inertia-viscous regime beyond a critical minimum value of the neck radius. The phenomenon of interface overturning is fundamentally related to the magnitude of drop viscosity. The variation of overturning angle as a function of drop viscosity is computed and a critical value of Ohnesorge number is obtained beyond which overturning ceases. Increasing the inertia of drop liquid transforms the system from a periodically dripping regime to a quasiperiodic regime and finally it culminates into an elongated liquid jet. Another interesting transition from dripping to jetting regime is demonstrated by varying the viscosity of the ambient medium. The breakup of jets in Rayleigh mode is explored and the breakup length obtained from our computations shows excellent agreement with the theoretical predictions owing to Rayleigh's analysis. The ambient medium is entrained as the jet moves downstream with the creation of a vortical structure just outside the jet signifying increased participation of the ambient medium in the dynamics of jet breakup at higher inflow rates.