The surge is a typical aerodynamic instability phenomenon in the compression system, which can lead to serious consequences such as engine performance degradation and structural damage. A deep understanding of the surge process can support the development of a compressor with a wider operating range. In this paper, an experimental study was carried out and high-responding pressure sensors were used to obtain the aerodynamic instability process and the post-surge characteristics of an axial–centrifugal compressor at design and off-design speeds. The evolution of the flow field and instability behavior before and after the surge were analyzed. The results showed that the inlet temperature change can reflect the aerodynamic instability to some extent, and as the operating condition moves from the choke to surge boundary, the inlet temperature undergoes a sudden increase at a certain condition and further increases with the decrease in mass flow rate. At the design speed, the instability of the combined compressor featured a deep surge with an obvious rotating stall behavior before its inception, and the amplitude of the stall cell was gradually enhanced, finally leading to the surge. At the off-design speed, affected by the stage mismatching, the axial stage mainly worked near the unstable operating condition. Therefore, the compressor successively experiences two modes of mild surge and deep surge, and the rotating stall can also be observed during the surge cycle.