We propose a novel integrated sensing and communication (ISAC) system that leverages sensing to assist communication, ensuring fast initial access, seamless user tracking, and uninterrupted communication for millimeter wave (mmWave) wideband systems. True-time-delayers (TTDs) are utilized to generate frequency-dependent radar rainbow beams by controlling the beam squint effect. These beams cover users across the entire angular space simultaneously for fast beam training using just one orthogonal frequency-division multiplexing (OFDM) symbol. Three detection and estimation schemes are proposed based on radar rainbow beams for estimation of the users' angles, distances, and velocities, which are then exploited for communication beamformer design. The first proposed scheme utilizes a single-antenna radar receiver and one set of rainbow beams, but may cause a Doppler ambiguity. To tackle this limitation, two additional schemes are introduced, utilizing two sets of rainbow beams and a multi-antenna receiver, respectively. Furthermore, the proposed detection and estimation schemes are extended to realize user tracking by choosing different subsets of OFDM subcarriers. This approach eliminates the need to switch phase shifters and TTDs, which are typically necessary in existing tracking technologies, thereby reducing the demands on the control circurity. Simulation results reveal the effectiveness of the proposed rainbow beam-based training and tracking methods for mobile users. Notably, the scheme employing a multi-antenna radar receiver can accurately estimate the channel parameters and can support communication rates comparable to those achieved with perfect channel information.