We experimentally study the chaotic behaviors in a compact all-fiber erbium-doped fiber ring laser (ED-FRL) with an added Mach-Zehnder interferometer (MZI) by using a phase-modulation method. A piezo-electric ceramic transducer (PZT) is incorporated in the MZI to introduce single-frequency phase modulation. The coexistence of intermittency and period-doubling bifurcation routes to chaos in the EDFRL system is observed by adjusting the modulation frequency or the phase modulation depth in the experiment. In addition, the EDFRL presents irregular multi-longitudinal-mode oscillation with a definite linear polarization when operating at intensity chaos state.