The coming era of gravitational wave (GW) astronomy, enabled by space-based laser interferometric detectors, heralds unprecedented opportunities to study compact binaries, in particular double white dwarfs (DWDs), double neutron stars (DNSs), and binary black holes, through their GW and electromagnetic observations, providing important insights into binary evolution, NS physics, and the overarching architecture of the Universe. In this paper, we provide an overview of our recent progress in the scientific objectives and data analysis relevant to these compact binaries within the scope of the space-based missions, emphasizing the multi-messenger observations of DWDs and DNSs in our Galaxy that synergize the capabilities of TianQin, LISA, and current and future optical and radio telescopes, and the promising multi-wavelength (dual-line) observations of the low-frequency GWs due to the binary motion of a DNS and the high-frequency GWs due to the spin of the aspherical NS component. We also briefly discuss the challenge of data analysis for the stellar-mass binary black holes for space-based detectors and present our preliminary solution, which involves both algorithmic innovations and hardware acceleration driven by graphics processing units (GPUs).