Multi-wavelength lasers and transverse-mode-switchable lasers are expected to find applications in Wavelength Division Multiplexing (WDM) and Mode Division Multiplexing (MDM) systems. High-order transverse modes play a crucial role in the generation of cylindrical vector beams and vortex beams, making them suitable for applications such as micro-particle manipulation, quantum information, and laser material processing. While in solid-state lasers, specific transverse modes can be excited through phase and amplitude modulation, the entire laser system is relatively underdeveloped, with limited mode scalability. Therefore, the all-fiber structure of multi-wavelength and transverse-mode-switchable fiber lasers has sparked significant interest among scientists. Various technological approaches have been proposed to achieve multi-wavelength or transverse mode outputs from fiber lasers. However, there has been limited reporting on fiber lasers capable of simultaneously operating at multiple wavelengths while generating switchable transverse modes. Reported multi-wavelength and transverse mode-switchable fiber lasers often employ gain media consisting of traditional doped fibers, where the competitive advantage of the fundamental mode within the resonant cavity is significantly greater than that of higher-order modes. Consequently, these fiber lasers struggle to maintain stable and efficient output of higher-order modes. In this study, a Ring-Core Yb-Doped Fiber (RCYDF) was designed and fabricated. The unique structure of the ring-shaped doping region aligns well with the dual-peak spatial electromagnetic field distribution of the LP₁₁ mode, prioritizing the gain acquisition for the LP₁₁ mode within the gain fiber. This design facilitates stable oscillation and efficient output of the LP₁₁ mode laser. Few-Mode Fiber Bragg Gratings (FMFBGs) serve not only as ideal wavelength-selective elements but also as components for achieving transverse mode switching. Utilizing a pair of FMFBGs as the laser resonant cavity and the fabricated RCYDF as the laser gain medium, a multi-wavelength and transverse-mode-switchable fiber laser was demonstrated. By simply adjusting the polarization controller placed on the RCYDF, stable laser oscillation at both single and dual wavelengths can be achieved. When operating in a single-wavelength state, two transverse modes, namely LP₀₁ and LP₁₁ modes, can be switched. The 3$\mathrm{d}\mathrm{B}$ linewidths for both modes are less than 0.08$\mathrm{n}\mathrm{m}$, with a Side-Mode Suppression Ratio (SMSR) of 52.2$\mathrm{d}\mathrm{B}$ for LP₀₁ mode and 46.5$\mathrm{d}\mathrm{B}$ for LP₁₁ mode. The output spectra were monitored every 10 minutes over a total duration of 60 minutes. Fluctuations in the central wavelength and optical intensity of the two laser modes were observed to be 0.01 nm and 1 dB, respectively. No significant changes were observed in the spectral shape. Furthermore, the laser thresholds for LP₀₁ and LP₁₁ modes are 372.51$\mathrm{m}\mathrm{W}$, and 482.51$\mathrm{m}\mathrm{W}$, respectively, with corresponding slope efficiencies of 34.34% and 49.09%. The higher slope efficiency of the LP₁₁ mode is attributed to the enhanced LP₁₁ mode resonance capability of the custom-designed RCYDF compared to traditional Yb-doped fibers, making it highly valuable for applications in fiber lasers targeting higher-order mode outputs. The proposed dual-wavelength and transverse-mode-switchable fiber laser offers advantages of simplicity, ease of control, and stable operation, presenting promising applications in MDM system and laser material processing.