In the era of precision studies in the flavor sector, it is paramount that the Standard Model (SM) of particle physics be tested further and the existence of physics beyond the standard model (BSM) or new physics (NP) be investigated via as many flavor avenues as possible. The flavor sector is a copious domain for testing the SM and for probing BSM physics. This sector has already seen many discoveries and has hinted at several new possibilities of BSM physics via flavor anomalies. The flavor beauty (b) and charm (c) quark sector is rich with several decay channels that have been experimentally observed and those that can possibly be measured at various current experimental facilities and future upgrades. In this work, it is intended that the several decay channels of the b and c hadrons be analyzed for new physics signatures in a phenomenological approach. These decay channels will be investigated within a model-independent and model-dependent approach containing new physics contributions in addition to standard model ones. In the model-dependent approach, leptoquarks have been considered to be strong candidates for explaining observed flavor anomalies and hence various types of leptoquarks will be considered for the proposed study. The Z' boson will also be considered in case of neutral current interactions. In the model-independent approach, the Standard Model Effective Field Theory (SMEFT) is a suitable framework for studies of new physics at a TeV scale or more. Hence, within these frameworks, various semileptonic decay channels involving the $\Lambda_b$ baryon, $B_s$ meson, $B_c$ meson, $D$ meson and a few others will be examined. Form factors for the hadronic transitions will be extracted from several models such as lattice QCD, covariant light front quark model, light cone sum rules, and others, wherever applicable. Wherever inputs are not available, calculations will be performed explicitly, including helicity amplitude calculations. Constraints on new physics contributions will be obtained from available experimental data in the corresponding sector. Numerical predictions of observables pertaining to these decay modes will then be made and the spectra of these observables in terms of the momentum transfer squared will also be evaluated. The results will then be compared with available experimental data, if any, to make appropriate conclusions. This study is significant as it will provide additional details on the fundamentals of particle physics, constrain and test possible NP, and complement other theoretical and experimental studies in the flavor sector.