This paper proposes a reinforcement learning-aided channel estimator for time-varying multi-input multi-output systems. The basic concept of the proposed channel estimator is the selection of the detected data symbol in the data-aided channel estimation. To achieve the selection successfully, we first formulate an optimization problem to minimize the data-aided channel estimation error. However, in time-varying channels, the optimal solution is difficult to derive because of its computational complexity and the time-varying nature of the channel. To address these difficulties, we consider a sequential selection for the detected symbols and a refinement for the selected symbols. A Markov decision process is formulated for sequential selection, and a reinforcement learning algorithm that efficiently computes the optimal policy is proposed with state element refinement. Simulation results demonstrate that the proposed channel estimator outperforms conventional channel estimators by efficiently capturing the variation of the channels.