We consider the {\it composite optimization problems} under convex and nonconvex settings. For the convex case, the {\it locally Lipschitz} condition is imposed on the gradient of the differentiable convex term. The classical {\it proximal gradient method} will be studied with our novel {\it enhanced adaptive} stepsize selection. To obtain the convergence of the proposed algorithm, we establish a sufficient decrease-type inequality associated with our new stepsize choice. This allows us to demonstrate the descent of the objective value from some fixed iteration and yields the sublinear convergence rate of the new method. Especially, in the case of locally strong convexity of the smooth term, our algorithm converges Q-linearly. When the gradient of the smooth term is globally Lipschitz, our method is extended to a wide class of {\it nonconvex} composite optimization problems where the smooth term can be convex, concave, indefinite quadratic or fractional (with an indefinite quadratic numerator and a positive affine denominator) functions. The experiments for our new method are conducted for seven practical problems with a large number of randomly generated data from small to large sizes. The superior efficiency of our novel proximal gradient algorithms is demonstrated by numerical results (evaluated by performance profile) in comparison with the other state-of-the-art methods.