In this paper, we propose new adaptive proximal gradient algorithms to solve multiobjective optimization problems, where each objective function is the sum of a differentiable function and a proper, closed, convex function. Utilizing the local behavior of the differentiable terms we propose new adaptive ways to select stepsizes used in proximal gradient scheme. In particular, the adaptive stepsizes are used in proximal subproblems whose optimal solution define the next iterates. We provide the convergence results for both nonconvex and convex cases of the considered problems. In particular, for a large class of nonconvex multiobjective composite optimization problems (where the differentiable terms are \textit{Quacavex} including convex, concave, indefinite quadratic, or indefinite quadratic over positive linear functions), we show that any cluster point of the iterates is Pareto stationary. If all objectives are convex, we prove the convergence of the sequence of iterates to a weakly Pareto optimal solution of the considered problem. From a finite step, the standard theoretical convergence rates of the proposed algorithms are established at $\mathcal{O}(1/\sqrt{K})$; $\mathcal{O}(1/K)$ and Q-linear for the nonconvex; convex and strongly convex settings, respectively. Notably, multiobjective proximal gradient algorithms in the literature commonly use stepsizes chosen by line search backtracking procedures or/and fixed stepsizes that requires the knowledge of the global Lipschitz gradient constant. The empirical performance is evaluated through practical models including robust optimization and supervised feature selection for benchmark datasets. Experimental results demonstrate that our proposed algorithm significantly outperforms existing methods, especially in terms of processing time.