In this paper, we address task and motion planning (TAMP) which is an important yet challenging robotics problem. It is known to suffer from the high combinatorial complexity of discrete search, often requiring a large number of geometric planning calls. We build upon recent works in TAMP by taking advantage of learning methods to provide action feasibility information as a heuristic to the symbolic planner, thus guiding it to a geometrically feasible solution and reducing geometric planning time. We propose \textbf{AGFP-Net}, a multi-task neural network predicting not only action feasibility, but also the feasibility of a set of grasp types. We also propose an improved feasibility-informed TAMP algorithm capable of solving more complex problems, and handling goals which are not fully specified. Comparative results obtained on different problems of varying complexity show that our method is able to greatly reduce task and motion planning time.