The effects of colored noise on stochastic resonance (SR) in a bistable system driven by multiplicative colored noise and additive white noise and a periodic signal are studied by using the unified colored noise approximation and the theory of signal-to-noise ratio (SNR) in the adiabatic limit. In the case of no correlations between noises, there is an optimal noise intensities ratio R at which SNR is a maximum that identifies the characteristics of the SR when the correlation time tau of the multiplicative colored noise is small. However, when tau is increased, a second optimal value of R appears, and two peaks appear in the SNR simultaneously. In the case of correlations between noises, the SNR is not only dependent on the correlation time tau, but also on the intensity of correlations between noises. Moreover, the double peak phenomenon can also appear as tau is increased in certain situations.