We perform density functional theory calculation to investigate the structural and electronic properties of various two-dimensional transition metal dichalcogenides, MX2 (M = Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, or W, and X =S or Se), and their strain-induced phase transitions. We evaluate the relative stability and the activation barrier between the octahedral-T and the trigonal-H phases of each MX2. It is found that the equilibrium and phase transition characteristics of MX2 can be classified by the group to which its metal element M belongs in the periodic table. MX2 with M in the group 4 (Ti, Zr, or Hf), forms an octahedral-T phase, while that with an M in the group 6 (Cr, Mo, or W) does a trigonal-H phase. On the other hand, MX2 with M in the group 5 (V, Nb, or Ta), which is in-between the groups 4 and 6, may form either phase with a similar stability. It is also found that their electronic structures are strongly correlated to the structural configurations: mostly metallic in the T phase, while semiconducting in the H phase, although there are some exceptions. We also explore the effects of an applied stress and find for some MX2 materials that the resultant strain, either tensile or compressive, may induce a structural phase transition by reducing the transition energy barrier, which is, in some cases, accompanied by its metal-insulator transition.