This is the first of two papers reporting measurements from a program to determine the Hubble constant to ~5% precision from a refurbished distance ladder. We present new observations of 110 Cepheid variables in the host galaxies of two recent Type Ia supernovae (SNe Ia), NGC 1309 and NGC 3021, using the Advanced Camera for Surveys on the Hubble Space Telescope (HST). We also present new observations of the hosts previously observed with HST whose SNe Ia provide the most precise luminosity calibrations: SN 1994ae in NGC 3370, SN 1998aq in NGC 3982, SN 1990N in NGC 4639, and SN 1981B in NGC 4536, as well as the maser host, NGC 4258. Increasing the interval between observations enabled the discovery of new, longer-period Cepheids, including 57 with P>60 days, which extend these period-luminosity (P-L) relations. We present 93 measurements of the metallicity parameter, 12 + log[O/H], measured from H II regions in the vicinity of the Cepheids and show these are consistent with solar metallicity. We find the slope of the seven dereddened P-L relations to be consistent with that of the Large Magellanic Cloud Cepheids and with parallax measurements of Galactic Cepheids, and we address the implications for the Hubble constant. We also present multi-band light curves of SN 2002fk (in NGC 1309) and SN 1995al (in NGC 3021) which may be used to calibrate their luminosities. In the second paper, we present observations of the Cepheids in the H band obtained with the Near-Infrared Camera and Multi-Object Spectrometer on HST, further mitigating systematic errors along the distance ladder resulting from dust and chemical variations. The quality and homogeneity of these SN and Cepheid data provide the basis for a more precise determination of the Hubble constant.