We present a search for the standard model Higgs boson produced in association with a $Z$ boson, using up to 7.9 fb$^{-1}$ of integrated luminosity from $p\bar{p}$ collisions collected with the CDF II detector. We utilize several novel techniques, including multivariate lepton selection, multivariate trigger parametrization, and a multi-stage signal discriminant consisting of specialized functions trained to distinguish individual backgrounds. By increasing acceptance and enhancing signal discrimination, these techniques have significantly improved the sensitivity of the analysis above what was expected from a larger dataset alone. We observe no significant evidence for a signal, and we set limits on the $ZH$ production cross section. For a Higgs boson with mass 115 GeV/$c^2$, we expect (observe) a limit of 3.9 (4.8) times the standard model predicted value, at the 95% credibility level.