A generally weighted moving average (GWMA) control chart for monitoring Poisson observations is introduced. It is showed that the Poisson GWMA chart outperforms other control charts, especially when the process shift is small. Furthermore, the Poisson GWMA chart provides a generalized model for which the Poisson EWMA and Shewhart c charts are the special cases. A novel control chart called “Poisson double GWMA” (PDGWMA) chart is also presented for monitoring Poisson-distributed processes. The PDGWMA chart is more sensitive than other control charts in detecting an out-of-control signal in most of situations, particularly in the cases of downward process shifts. Consequently, not only the quality deterioration but also the quality improvement can be detected agilely. Some approaches have been proposed to modify EWMA charts with fast initial response (FIR) features. In this research, we employ these approaches in Poisson GWMA charts to agilely detect the process shifts caused by the start-up quality problems. It is showed that the PDGWMA chart is the first choice in detecting moderate or small shifts especially when the shifts are downward, and the PGWMA chart with adjusted time-varying control limits performs excellently in detecting great process shifts during the initial stage.