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This study proposes a new multilevel control of a quadrotor with dynamic uncertainties and time-varying external disturbances. The quadrotor model is partitioned into three subsystems: the vertical position, the horizontal position and the rotational subsystems. First, a double loop integral fast terminal sliding mode control with an adaptive estimator for disturbances' upper-bounds (ADIFTSMC) is proposed for the altitude subsystem to ensure that the quadrotor reaches the desired height. Secondly, a radial basis function neural network backstepping controller (RBFNNBC) is applied to the horizontal subsystem. Finally, by combining a finite time exact disturbance observer with backstepping nonsingular fast terminal sliding mode control (FDOBNFTSMC), the rotational angles converge to the reference angles in the presence of the time-varying disturbances. Furthermore, a Lyapunov stability analysis is used to prove that the tracking errors converge to a small neighborhood of the origin. Numerical simulations illustrate the feasibility of the compound control structure.