$beta$-decay spectroscopy provides valuable information on exotic nuclei and a stringent test for nuclear theories beyond the stability line. To search for new $beta$-delayed protons and $gamma$ rays of $^{25}$Si to investigate the properties of $^{25}$Al excited states. $^{25}$Si $beta$ decays were measured by using the Gaseous Detector with Germanium Tagging system at the National Superconducting Cyclotron Laboratory. The protons and $gamma$ rays emitted in the decay were detected simultaneously. A Monte Carlo method was used to model the Doppler broadening of $^{24}$Mg $gamma$-ray lines caused by nuclear recoil from proton emission. Shell-model calculations using two newly developed universal textit{sd}-shell Hamiltonians, USDC and USDI, were performed. The most precise $^{25}$Si half-life to date has been determined. A new proton branch at 724(4)~keV and new proton-$gamma$-ray coincidences have been identified. Three $^{24}$Mg $gamma$-ray lines and eight $^{25}$Al $gamma$-ray lines are observed for the first time in $^{25}$Si decay. The first measurement of the $^{25}$Si $beta$-delayed $gamma$ ray intensities through the $^{25}$Al unbound states is reported. All the bound states of $^{25}$Al are observed to be populated in the $beta$ decay of $^{25}$Si. Several inconsistencies between the previous measurements have been resolved, and new information on the $^{25}$Al level scheme is provided. An enhanced decay scheme has been constructed and compared to the mirror decay of $^{25}$Na and the shell-model calculations. The measured excitation energies, $gamma$-ray and proton branchings, log~$ft$ values, and Gamow-Teller transition strengths for the states of $^{25}$Al populated in the $beta$ decay of $^{25}$Si are in good agreement with the shell-model calculations, offering gratifyingly consistent insights into the fine nuclear structure of $^{25}$Al.
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