Unidentified Aerial Phenomena (UAP) have resisted explanation and have received little formal scientific attention for 75 years. A primary objective of the Galileo Project is to build an integrated software and instrumentation system designed to conduct a multimodal census of aerial phenomena and to recognize anomalies. Here we present key motivations for the study of UAP and address historical objections to this research. We describe an approach for highlighting outlier events in the high-dimensional parameter space of our census measurements. We provide a detailed roadmap for deciding measurement requirements, as well as a science traceability matrix (STM) for connecting sought-after physical parameters to observables and instrument requirements. We also discuss potential strategies for deciding where to locate instruments for development, testing, and final deployment. Our instrument package is multimodal and multispectral, consisting of (1) wide-field cameras in multiple bands for targeting and tracking of aerial objects and deriving their positions and kinematics using triangulation; (2) narrow-field instruments including cameras for characterizing morphology, spectra, polarimetry, and photometry; (3) passive multistatic arrays of antennas and receivers for radar-derived range and kinematics; (4) radio spectrum analyzers to measure radio and microwave emissions; (5) microphones for sampling acoustic emissions in the infrasonic through ultrasonic frequency bands; and (6) environmental sensors for characterizing ambient conditions (temperature, pressure, humidity, and wind velocity), as well as quasistatic electric and magnetic fields, and energetic particles. The use of multispectral instruments and multiple sensor modalities will help to ensure that artifacts are recognized and that true detections are corroborated and verifiable. Data processing pipelines are being developed that apply state-of-the-art techniques for multi-sensor data fusion, hypothesis tracking, semi-supervised classification, and outlier detection.