We have investigated in detail the continuous-wave (cw) and mode-locked performance of a diode-pumped Cr:Nd:GSGG laser. State-of-the-art single-mode and multimode laser diodes around 665 nm were used as pump sources. In cw operation, we have demonstrated lasing thresholds as low as 14 mW, slope efficiencies as high as 23.4%, and output powers up to 738 mW. The free running emission wavelength was 1061 nm. Lasing could also be obtained at 1051, 1058, 1065, 1068, 1072, 1103, and 1111 nm lines. A saturable Bragg reflector was used to initiate and sustain mode-locking where the Cr:Nd:GSGG laser produced 6-ps-long pulses around 1061 nm with an average power of 160 mW. The repetition rate was 142.65 MHz, resulting in pulse energies of 1.1 nJ and peak powers of 175 W. An off-surface optical axis quartz birefringent filter (BRF) was inserted inside the laser cavity at Brewster's angle to obtain two-color cw and mode-locked laser operation at the 1051 and 1058 nm and 1058 and 1061 transition pairs, resulting in cw powers up to 60 mW and cw mode-locked average powers up to 45 mW. Unlike many other methods applied for two-color mode-locked laser operation, usage of the BRF enabled regulation of the ratio of the power in each line by fine adjustment of its rotation angle. The method could potentially be used for other gain media as well, which could simplify development of multicolor solid-state laser systems.