Two-dimensional (2D) transition metal carbides known as MXenes belong to a new branch of 2D material family, and their fundamental properties vary with their compositions and surface functionalizations. In this study, the structural and ideal mechanical properties of M2C-type MXenes and their functionalized M2CT2 MXenes (M = Ti, Zr, Hf; T = O, F, OH) were systematically examined via first-principles methods. The stress-strain curves of the MXenes under homogenous biaxial and uniaxial tension are identified, and the fundamental quantities (e.g., Young's modulus, in-plane stiffness, and Poisson's ratio) are addressed. With significantly higher strength and extended critical strains, the M2CO2 MXenes exhibit optimal flexibility when compared with that of M2C, M2CF2, and M2C(OH)(2). Additionally, Hf2CT2 exhibits optimal tensile performance under uniaxial or biaxial tension when compared to that of Ti2CT2 and Zr2CT2. The Young's modulus, in-plane stiffness, and Poisson's ratio of MXenes with different surface functionalization increase in a sequence corresponding to OH < F < O. Furthermore, the effects of vacancy on the mechanical properties of MXenes are further explored and indicate that vacancy can significantly weaken the tensile properties of MXenes that are considered. Moreover, vacancy also results in a certain anisotropy of stress along armchair and zigzag directions even under the biaxial tension condition.