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We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by the summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient value of the original objective and constraint functions. Either exact or approximate subproblem solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting, which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where 1) the objective is a stochastic or finite-sum function, and 2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function-constrained problems where we show complexities similar to the proximal gradient method.