Unlike most control systems, kinematic uncertainty is present in robot control systems in addition to dynamic uncertainty. The use of different types of external sensors in various configurations also results in different sensory transformation or Jacobian matrices and thus leads to different kinematic models. Currently, there is no systematic theoretical framework in developing data-driven neural network (NN) learning and control methods for task-space tracking control of robots with unknown kinematics and dynamics. The existing NN controllers are limited to either dynamic control or kinematic control without considering the interaction between the inner control loop and the outer control loop. In this paper, a NN based data driven offline learning algorithm and an online learning controller are proposed, which are combined in a complementary way. The proposed task-space control algorithms can be implemented on robotic systems with closed control architecture by considering the interaction with the inner control loop. Theoretical analyses are presented to show the stability of the systems and experimental results are presented to illustrate the performance of the proposed learning algorithms.