The See Change survey was designed to make $z>1$ cosmological measurements by efficiently discovering high-redshift Type Ia supernovae (SNe Ia) and improving cluster mass measurements through weak lensing. This survey observed twelve galaxy clusters with the Hubble Space Telescope spanning the redshift range $z=1.13$ to $1.75$, discovering 57 likely transients and 27 likely SNe Ia at $z\sim 0.8-2.3$. As in similar previous surveys (Dawson et al. 2009), this proved to be a highly efficient use of HST for SN observations; the See Change survey additionally tested the feasibility of maintaining, or further increasing, the efficiency at yet higher redshifts, where we have less detailed information on the expected cluster masses and star-formation rates. We find that the resulting number of SNe Ia per orbit is a factor of $\sim 8$ higher than for a field search, and 45% of our orbits contained an active SN Ia within 22 rest-frame days of peak, with one of the clusters by itself yielding 6 of the SNe Ia. We present the survey design, pipeline, and SN discoveries. Novel features include fully blinded SN searches, the first random forest candidate classifier for undersampled IR data (with a 50% detection threshold within 0.05 magnitudes of human searchers), real-time forward-modeling photometry of candidates, and semi-automated photometric classifications and follow-up forecasts. We also describe the spectroscopic follow-up, instrumental in measuring host-galaxy redshifts. The cosmology analysis of our sample will be presented in a companion paper.