The growth dynamics of gold-silver core-shell (Au@Ag) nanoparticles are studied using in situ time-dependent second harmonic generation (SHG) and extinction spectroscopy to investigate the nanoparticle shell formation. The silver shell is grown by reduction of silver cations onto a 14 nm gold core using ascorbic acid in colloidal aqueous solution under varying reaction concentrations producing Au@Ag nanoparticles of final sizes ranging from 51 to 78 nm in diameter. The in situ extinction spectra show a rapid increase in intensity on the timescale of 5-6 s with blue shifting and narrowing of the plasmonic peak during the silver shell formation. The in situ SHG signals show an abrupt rise at early times of the reaction, followed by a time-dependent biexponential decrease, where the faster SHG lifetime corresponds to the timescale of the shell growth, and where the slower SHG lifetime is attributed to changes in the nanoparticle surface charge density. A large enhancement in the SHG signal at early stages of the reaction is caused by plasmonic hot spots due to the nanoparticle surface morphology, which becomes smoother as the reaction proceeds. The final extinction spectra are compared to finite-difference time-domain (FDTD) calculations, showing general agreement with experiment, where the plasmon peak red shifts and increases in spectral width as the silver shell thickness increases. These in situ SHG and extinction spectroscopy results, combined with FDTD calculations, help characterize the complicated processes involved in colloidal nanoparticle shell formation in real time for developing potential plasmon-enhanced nanomaterial applications.