We present ORCAst, a multi-stage, multi-arm network for Operational high-Resolution Current forecAsts over one week. Producing real-time nowcasts and forecasts of ocean surface currents is a challenging problem due to indirect or incomplete information from satellite remote sensing data. Entirely trained on real satellite data and in situ measurements from drifters, our model learns to forecast global ocean surface currents using various sources of ground truth observations in a multi-stage learning procedure. Our multi-arm encoder-decoder model architecture allows us to first predict sea surface height and geostrophic currents from larger quantities of nadir and SWOT altimetry data, before learning to predict ocean surface currents from much more sparse in situ measurements from drifters. Training our model on specific regions improves performance. Our model achieves stronger nowcast and forecast performance in predicting ocean surface currents than various state-of-the-art methods. SIGNIFICANCE STATEMENT: Our study introduces a novel neural network designed to produce high-resolution, real-time 7-day forecasts of ocean surface currents. Accurate forecasting of ocean currents is important for ship routing, climate studies, and tracking of pollutants. This is a challenging problem, due to sparse in situ observations, indirect or incomplete satellite observations, as well as complex ocean dynamics. We develop a multi-stage training procedure to learn ocean currents by progressively refining forecasts with data of increasing quality. Our method only uses real satellite and in situ data, including the latest generation of satellite altimetry from the recently launched SWOT satellite, and does not depend on numerical simulations.