Nuclear size and shape are fundamental physical properties defined by the composition of neutrons and protons within the nucleus. The precise investigation of both properties provides benchmarks for nuclear models and helps to understand the nucleon-nucleon interaction. The nuclear charge radius and the nuclear magnetic dipole and electric quadrupole moments can be probed by the well-known electromagnetic interaction and extracted from atomic spectra. Collinear laser spectroscopy (CLS) is a versatile tool to measure and resolve fine structure and hyperfine structure required to extract the nuclear parameters. It is an in-flight spectroscopy method and therefore very well suited for short-lived isotopes and isomeric states. A new online collinear laser spectroscopy setup, the Argonne Tandem Hall Laser Beamline for Atom and Ion Spectroscopy (ATLANTIS) has been commissioned in this work with first online beams of 110,112−116,118Pd delivered from the Californium Rare Isotope Breader Upgrade (CARIBU). The required high sensitivity for these investigations of palladium isotopes with very low production yields was achieved by a new radio-frequency quadrupole cooler-and-buncher with unprecedented long accumulation times of up to 30 s. A new charge-exchange cell, designed specifically for the charge exchange with magnesium, provided unexpected high neutralization efficiency with almost undisturbed and symmetric resonance profiles. Using the new setup, spectra of all stable and the neutron-rich 112−116,118Pd isotopes were obtained, the spins of 113,115Pd confirmed, their magnetic dipole moments and the electric quadrupole moment of 113Pd determined, and the trend of the differential nuclear charge radii along the chain of measured isotopes established. The results indicate a complex nuclear structure and are compared to other isotopic chains in the Pd region and to axial Fayans EDF and triaxial Skyrme EDF calculations. The good agreement with the EDF calculations allowing triaxiality, supports the possible existence of triaxial ground states in the mid-shell region between Z=26 and Z=50 and around N=60. Moreover, CLS measurements were performed at the COLLAPS experiment at ISOLDE/CERN on semi-magic lead isotopes. The differential mean-square nuclear charge radii and nuclear moments of 187−208Pb and of the isomeric states of the odd 187−203Pb isotopes were determined. A comparison to Fayans and ab initio calculations show an overall good greement.