Geometric partial differential equations, such as mean-curvature flow and surface diffusion, are challenging to solve numerically due to their strong non-linearity and stiffness, when solved explicitly. Solving these high-order PDEs using explicit methods would require very small time steps to achieve stability, whereas using implicit methods would result in complex nonlinear systems of equations that are expensive to solve. In addition, accurate spatial discretizations of these equations pose challenges in their own rights, especially on triangulated surfaces. We propose new methods for mean curvature flow and surface diffusion using triangulated surfaces. Our method uses a weighted least-squares approximation for improved accuracy and stability, and semi-implicit schemes for time integration for larger time steps and higher efficiency. If mesh element quality is initially poor, or becomes poor through evolution under mean curvature flow or surface diffusion, we utilize mesh adaptivity to improve mesh quality and proceed further in evolution. Numerical experiments and comparisons demonstrate that our method can achieve second-order accuracy for both mean-curvature flow and surface diffusion, while being much more accurate and stable than using explicit schemes or alternative methods.