Current material extrusion systems can produce complex parts but lack instrumentation for observability and control. To investigate methods for observing the material extrusion process, a printer is instrumented to examine the dependency chain from the motor shaft torque to the infeed load and finally the melt pressure and temperature. The transient rheological and thermal behavior of the material extrusion process and the effect of volumetric flow rate, nozzle orifice diameter, and temperature setpoint on the pressure estimate from each point in the dependency chain are reported. The work also presents pressure predictions from COMSOL Multiphysics non-isothermal flow simulations and an analytical (Poiseuille) model. The pressure estimated by the motor shaft torque is greater than the downstream pressure estimated by the infeed load, which is greater than the downstream melt pressure in the hot end. In other words, both the torque sensor and the infeed load significantly overpredict the melt pressure. Significant variations in the pressures are also observed and explained. The findings demonstrate low and high frequency variation in the process, which can be attributed to gear eccentricity and teeth-to-filament engagement. The melt pressure variation is also observed to increase significantly at lower temperature set-points and higher flow rates, both of which reduce the melt temperature and thereby increase the viscosity. The increase in viscosity tends to reduce the viscous damping such that the variations in the filament infeed are transmitted through the hot end to the extrudate.