We investigate and report the broadband flat near-infrared (NIR)/mid-infrared (MIR) emissions of Tm3+–Ho3+ co-doped, and Tm3+–Ho3+–Yb3+ tri-doped SiO2–ZnO–K2O–La2O3 zinc silicate glasses under excitation of 808 and 980 nm laser diode (LD). Broadband relatively flat MIR emissions of Tm3+–Ho3+ co-doped, and Tm3+–Ho3+–Yb3+ tri-doped with a full-width at half-maximum (FWHM) of about 420, and 390 nm corresponding to commercial of 808, 980 nm LD excitations were identified. The role of Yb3+ in enhancing the intensity of the NIR/MIR emissions of Tm3+–Ho3+ co-doped, and Tm3+–Ho3+–Yb3+ tri-doped was also investigated. The GTm(λ), GHoλ) gain coefficient spectra of Tm3+ single-doped, and Ho3+ single-doped were determined respectively. Besides, the mechanisms for cross-relaxation (CR), energy transfer (ET), cooperative ET (CET) processes between Tm3+, Ho3+, and Yb3+ are proposed and discussed.