We present stacked thermal Sunyaev-Zel’dovich (tSZ) effect measurements for three samples of galaxy groups and clusters: those traced by the Dark Energy Spectroscopic Instrument Data Release 1 (DESI DR1) luminous red galaxies (LRG) and the DESI DR1 Bright Galaxy Sample (BGS), and an eROMaPPer optically-selected sample from the DESI Legacy Imaging Survey. We use the latest Atacama Cosmology Telescope DR6 (ACT)+Planck component-separated internal linear combination (ILC) Compton-y maps and ACT+Planck coadded 90, 150, and 220 GHz temperature maps to extract the tSZ signal within a ~2 disk aperture for sources binned by luminosity, richness, or mass. We measure the average tSZ signal with high statistical significance, with signal-to-noise ratios surpassing 38 for LRG, 27 for BGS, and 39 for the eROMaPPer sample using the 90 GHz ACT DR6+Planck map. We conduct a detailed study of systematics and foregrounds such as dust and cosmic infrared background (CIB) contamination, which remain a core challenge for tSZ analysis. For the LRG and BGS samples, we find that dust and radio source emission dominate the tSZ signal at scales near and below the disk aperture radius. Large-scale (R > 4) contamination from the CIB is less significant. We mitigate these contaminants to isolate the tSZ signal and use a combination of simulated and real measurements to develop Compton-y-optical depth scaling relations to infer optical depths, which are found to be in agreement with values measured using the pairwise kinematic SZ effect for the same tracer samples. The scaling relation for the eROMaPPer sample is the first such relationship to be derived directly from SZ measurements.