To mitigate the effects of shadow fading and obstacle blocking, reconfigurable intelligent surface (RIS) has become a promising technology to improve the signal transmission quality of wireless communications by controlling the reconfigurable passive elements with less hardware cost and lower power consumption. However, accurate, low-latency and low-pilot-overhead channel state information (CSI) acquisition remains a considerable challenge in RIS-assisted systems due to the large number of RIS passive elements. In this paper, we propose a three-stage joint channel decomposition and prediction framework to require CSI. The proposed framework exploits the two-timescale property that the base station (BS)-RIS channel is quasi-static and the RIS-user equipment (UE) channel is fast time-varying. Specifically, in the first stage, we use the full-duplex technique to estimate the channel between a BS's specific antenna and the RIS, addressing the critical scaling ambiguity problem in the channel decomposition. We then design a novel deep neural network, namely, the sparse-connected long short-term memory (SCLSTM), and propose a SCLSTM-based algorithm in the second and third stages, respectively. The algorithm can simultaneously decompose the BS-RIS channel and RIS-UE channel from the cascaded channel and capture the temporal relationship of the RIS-UE channel for prediction. Simulation results show that our proposed framework has lower pilot overhead than the traditional channel estimation algorithms, and the proposed SCLSTM-based algorithm can also achieve more accurate CSI acquisition robustly and effectively.