The thesis is split into three parts: In the first part we describe the Geometric Trinity of Gravity, i.e. the three alternative formulations of gravitational interactions. General Relativity uses the curvature of spacetime to describe gravity. However, there are two other alternative but dynamically equivalent formulations: the Teleparallel theory of gravity, which suggests that gravity is mediated through the torsion of spacetime and the Symmetric Teleparallel gravity that assigns gravity to the non-metricity of spacetime. In addition, we discuss possible modifications in each case. In the second part, we use Lie and Noether symmetries of modified theories of gravity as a geometric criterion to classify them on those that are invariant under point transformations. Furthermore, we calculate the invariants of each symmetry and use them to reduce the dynamics of each system in order to find exact cosmological solutions. However, modified theories should also behave ``correctly'' at astrophysical scales too. That is why, in the last part, we use the notion of the maximum turnaround radius of a structure as a stability criterion to test theories of gravity. Specifically, we derive a general formula for the maximum turnaround radius, which denotes the maximum size that a structure can have, for all theories that respect the Einstein Equivalence Principle. Finally, we apply this formula to the Brans-Dicke and the f(R) theories and discuss the requirements for the stability of large scale structures in their framework
