In this work, buckling and vibration characteristics of two-dimensional functionally graded nanobeam of nonuniform thickness subjected to in-plane and thermal loads have been analyzed within the frame work of Timoshenko beam theory. The beam is tapered by linear variation in thickness along the length. The temperature dependent material properties of the beam are varying along thickness and length as per a power-law distribution and exponential function, respectively. The analysis has been presented employing Eringen's nonlocal theory to incorporate the size effect. Hamilton's energy principle has been used to formulate the governing equations of motion. These resulting equations have been solved via generalized differential quadrature method for three combinations of clamped and simply supported boundary conditions. The effect of in-plane load together with temperature variation, nonuniformity parameter, gradient indices, nonlocal parameter and slenderness ratio on the natural frequencies is illustrated for the first three modes of vibration. The critical buckling loads in compression have been computed by putting the frequencies equal to zero. A significant contribution of in-plane load on mechanical behavior of two-directional functionally graded nanobeam with nonuniform cross-section has been noticed. Results are in good accordance.