Classical Cepheids are the most popular distance indicators and tracers of young stellar populations. The key advantage is that they are bright and they can be easily identified in Local Group and Local Volume galaxies. Their evolutionary and pulsation properties depend on their chemical abundances. The main aim of this investigation is to perform a new and accurate abundance analysis of 20 calibrating Galactic Cepheids. We used high spectral resolution (R~40000-115000) and high S/N spectra (~400), covering the entire pulsation cycle. We focused our attention on plausible systematics that would affect the estimate of atmospheric parameters and elemental abundances along the pulsation cycle. We cleaned the line list by using atomic transition parameters based on laboratory measurements and by removing lines that are either blended or that display abundance variations along the pulsation cycle. The spectroscopic approach we developed brings forward small dispersions in the variation of the atmospheric parameters (sigma(Teff)~50K, sigma(logg)~0.2dex, and sigma(xi)~0.2km/s) as well as in the abundance of both iron (<~0.05dex) and alpha elements (<~0.10dex) over the entire pulsation cycle. We also provide new and accurate effective temperature templates by splitting the calibrating Cepheids into four different period bins, ranging from short to long periods. For each period bin, we performed an analytical fit with Fourier series providing theta=5040/Teff as a function of the pulsation phase. The current findings are a good viaticum for tracing the chemical enrichment of the Galactic thin disk by using classical Cepheids as a fundamental stepping stone for further investigations into the more metal-poor regime that is typical of Magellanic Cepheids.