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We introduce a new, very deep neutral hydrogen (HI) survey being performed with the Westerbork Synthesis Radio Telescope (WSRT). The Westerbork Hydrogen Accretion in LOcal GAlaxieS (HALOGAS) Survey is producing an archive of some of the most sensitive HI observations available, on the angular scales which are most useful for studying faint, diffuse gas in and around nearby galaxies. The survey data are being used to perform careful modeling of the galaxies, characterizing their gas content, morphology, and kinematics, with the primary goal of revealing the global characteristics of cold gas accretion onto spiral galaxies in the local Universe. In this paper, we describe the survey sample selection, the data acquisition, reduction, and analysis, and present the data products obtained during our pilot program, which consists of UGC 2082, NGC 672, NGC 925, and NGC 4565. The observations reveal a first glimpse of the picture that the full HALOGAS project aims to illuminate: the properties of accreting HI in different types of spirals, and across a range of galactic environments. None of the pilot survey galaxies hosts an HI halo of the scale of NGC 891, but all show varying indications of halo gas features. We compare the properties of detected features in the pilot survey galaxies with their global characteristics, and discuss similarities and differences with NGC 891 and NGC 2403.