We present the goals, strategy, and first results of the high-cadence Galactic plane survey using the Zwicky Transient Facility (ZTF). The goal of the survey is to unveil the Galactic population of short-period variable stars, including short-period binaries, and stellar pulsators with periods less than a few hours. Between 2018 June and 2019 January, we observed 64 ZTF fields resulting in 2990 deg² of high stellar density in the ZTF-r band along the Galactic plane. Each field was observed continuously for 1.5 to 6 h with a cadence of 40 sec. Most fields have between 200 and 400 observations obtained over 2–3  continuous nights. As part of this survey, we extract a total of ≈230 million individual objects with at least 80 epochs obtained during the high-cadence Galactic plane survey reaching an average depth of ZTF–r ≈ 20.5 mag. For four selected fields with 2–10 million individual objects per field, we calculate different variability statistics and find that ≈1–2  per cent of the objects are astrophysically variable over the observed period. We present a progress report on recent discoveries, including a new class of compact pulsators, the first members of a new class of Roche lobe filling hot subdwarf binaries as well as new ultracompact double white dwarfs and flaring stars. Finally, we present a sample of 12 new single-mode hot subdwarf B-star pulsators with pulsation amplitudes between ZTF–r = 20–76 mmag and pulsation periods between P = 5.8–16 min with a strong cluster of systems with periods ≈6 min. All of the data have now been released in either ZTF Data Release 3 or Data Release 4.