Quantum gravity effective action: corrections to classical metrics and observational tests

General relativity is non-renormalizable, meaning that we lack a complete quantum theory of gravity. However, working at energies below the Planck mass, which is the cutoff scale of quantum gravity, we can resort to the effective field theory approach, implemented here via the Barvinsky-Vilkovisky unique effective action. The latter has been used to derive quantum corrections to classical metrics, such as the Schwarzschild solution. In this thesis, we compute these corrections and extend the analysis to a static and electrically charged star modeled as a perfect fluid, considering distinct scenarios based on different applications of the perfect fluidity condition to the energy-momentum tensor components. Additionally, we explore gravastars and dark energy stars, proposed as compact objects alternative to classical Schwarzschild black holes, arguing that quantum-induced hairs in their metrics may allow us to experimentally distinguish between these objects. Finally, we examine gravitational lensing observables, namely the photon sphere radius and the deflection angle of bent light rays, and the gravitational redshift, to validate the predicted quantum corrections. These findings provide a framework for potential empirical tests to distinguish quantum-corrected metrics from classical ones, contributing to the broader understanding of quantum gravity phenomenology