Reconfigurable intelligent surface (RIS) has been recognized as an efficient solution to realize integrated sensing and communications (ISAC) by dynamically reconfiguring wireless channels, improving signal quality, and mitigating issues such as signal attenuation and blockages, especially in high-frequency millimeter-wave band. However, traditional sensing signal processing methods are only applicable for narrowband systems. Due to the wideband effects that ISAC signals significantly varies across different frequencies, sensing signal processing methods have to be newly designed in order to fulfill the potential of RIS-aided ISAC systems. To address this issue, in this paper, we propose a wideband signal processing approach for velocity and range estimation in RIS-aided ISAC systems, where we flatten the frequency property of ISAC signals through sub-band division. Specifically, we first employ a discrete Fourier transform (DFT)-based method to initialize the Doppler shift and time delay of the target. Then, we propose a sub-band division method, based on which we solve the maximum likelihood estimation problem by iteratively optimizing Doppler shift, time delay, as well as the frequency-domain amplitudes on multiple sub-bands. Simulation results demonstrate that the proposed wideband signal processing method significantly outperforms the traditional narrowband signal processing approaches in terms of velocity and range estimation accuracy, especially with high SNR and proper sub-band division.