In this paper, we make a comprehensive study on tailoring the spectrum and transverse mode of random fiber lasers (RFLs). By simply temperature tuning, the mode gain profile of RFL can be flexibly and precisely manipulated. The spectrum of laser output can be easily tailored in single-wavelength, dual-wavelength, and three-wavelength, respectively. Meanwhile, the operating transverse mode is also optional among LP01 mode, LP11 mode, and hybrid mode. The slope efficiency of 17.9% and 27.3% are obtained for LP11 mode and LP01 mode operation, respectively. Besides, the coherence control can be confirmed by making speckle contrast measurements. This high-efficiency RFL with the customizable spectrum and spatial mode would have unique applications in wavelength or mode division multiplexing systems, speckle-free imaging, secure communication, and information encryption.