The Laser Interferometer Space Antenna (LISA) will explore the yet-uncharted millihertz band of the gravitational-wave (GW) spectrum in between the very low frequencies probed by pulsar timing arrays and the kilohertz window accessible to ground-based observatories. LISA's adoption is scheduled for early 2024 and data analysis methods are well underway, given the novelty and complexity of the expected data. The GalaxyFIT project aims at designing methods that simultaneously measure and describe the LISA noise, the gravitational-wave signals from thousands of sources, plus gravitational-wave backgrounds from astrophysical and cosmological sources. Integrated methods starting from non-ideal interferometric data all the way to data quality assessments of partially resolved sources are a necessity for LISA to reach its science objectives. GalaxyFIT will focus on ultracompact binary sources in the Milky Way where thousands of sources will be individually resolved and many more will result in an unresolved background, dominating the instrument noise and hampering the detection of a cosmological gravitational wave background. Specifically, GalaxyFIT will establish improved methods for detection and characterization of sources in the presence of non-ideal instrumental noise, and assess the impacts of different data quality cuts on astrophysical inference (measurement of Galactic structure parameters and features in the distributions of the binaries) and the measurement of a cosmological background. GalaxyFIT gathers experts on the LISA instrument, data analysis and astrophysical interpretation and aims to address many of the important data analysis and scientific goals that France has assumed for the LISA Mission, and pave the way for maximised scientific output with the LISA data. GalaxyFIT will also train a new generation of scientists towards this goal.