We report on a spectroscopic determination of the stellar parameters and chemical abundances for the parent star of the transiting planet TrES-1. Based on a detailed analysis of iron lines in our Keck and Hobby-Eberly Telescope spectra, we derive Teff=5250+/-75 K, logg=4.6+/-0.2, and [Fe/H]=0.00+/-0.09. By measuring the Ca II activity indicator and by putting useful upper limits on the Li abundance, we constrain the age of TrES-1 to be 2.5+/-1.5 Gyr. By comparing theoretical stellar evolution models with the observational parameters, we obtain M*=0.89+/-0.05 Msolar and R*=0.83+/-0.05 Rsolar. Our improved estimates of the stellar parameters are utilized in a new analysis of the transit photometry of TrES-1 to derive a mass Mp=(0.76+/-0.05)MJ, a radius Rp=1.04+0.08-0.05RJ, and an inclination i=89.5+0.5-1.3 deg. The improved planetary mass and radius estimates provide the grounds for new crucial tests of theoretical models of evolution and evaporation of irradiated extrasolar giant planets.