The properties of strongly correlated electron systems, in particular the correlated Mott insulators commonly encountered among transition metal oxides, are at the forefront of current research in condensed matter physics. One of the central problems in the field, particularly due to its relevance to photoemission experiments, is the behaviour of a charge injected into a Mott insulator which can couple to the ordered spin-orbital degrees of freedom to form a polaron. The questions of the existence and the dynamical properties of the ensuing quasiparticle state can be elucidated by means of Green's function calculations from effective models of the system. In this thesis we explore the theory of spin and orbital polarons, i.e., quasiparticles resulting from the charge coupling to magnetic or orbital long range order. To this end, we develop effective models for two structurally similar systems: the copper-oxide series of high temperature superconductors (or cuprates, modeled using the $t$-$J$ model and its extensions), and the copper-fluoride perovskite (KCuF3, modelled using the Kugel-Khomskii model). We then develop analytical and numerical methods for calculating single electron Green's functions by means of expansion around an ordered ground state, namely the Green's function variational approximation and the self-consistent Born approximation. Subsequently, we apply these methods to solve three related polaronic model systems: purely spin planar model based on cuprates, purely orbital planar model inspired by CuF2 planes of KCuF3, and the full three dimensional spin-orbital model for KCuF3 which has never been solved before. By comparing the results for the two cases with a single degree of freedom we demonstrate the differences between the spin and orbital interactions for the polaronic properties and draw general conclusions about the spin-orbital model. Further, we demonstrate a number of interesting effects encountered in the spin-orbital problem, such as dimensional interplay between orbitals and spins leading to polarons of predominantly orbital nature in the strong coupling regime; the orbital to spin polaron crossover under varying superexchange strength; or the importance of the Hund's exchange in the settling of the magnetic ground state. We conclude by discussing open problems and proposing possible routes of continuation of the present work.