In this thesis, we study inflationary dynamics, cosmic evolution and structure of hypothetical geometries. Firstly, we investigate the behavior of warm intermediate and logamediate inflationary models for flat isotropic and homogeneous universe in Einstein frame representation of f(R) gravity. In this scenario, we study the dynamics of strong and weak constant as well as generalized dissipative regimes. In both regimes, we discuss inflaton solution, slow-roll parameters, scalar and tensor power spectra, corresponding spectral indices as well as tensor-scalar ratio for Starobinsky inflationary model and determine their compatibility with Planck 2015 constraints. Secondly, we study the existence of Noether symmetry and associated conserved quantity of some isotropic as well as anisotropic universe models in f(R,T) gravity. The cyclic variable is introduced to construct exact solution of Bianchi I model. We also consider a generalized spacetime which corresponds to different anisotropic homogeneous universe models and scalar field model (quintessence and phantom) admitting minimal coupling with f(R,T) models. For these models, we formulate exact solutions without introducing cyclic variable. We investigate the behavior of some cosmological parameters using exact solutions through graphical analysis. Finally, we discuss wormhole solutions of static spherically symmetric spacetime via Noether symmetry approach in f(R) and f(R,T) theories. We formulate symmetry generators, associated conserved quantities and wormhole solutions for constant as well as variable red-shift functions. For perfect fluid, we evaluate an explicit form of generic function f(R) and also evaluate exact solution for f(R) power-law model. In f(R,T) gravity, we consider two f(R,T) models appreciating indirect curvaturematter coupling and formulate solutions for both dust as well as perfect fluids. We study the behavior of null/weak energy conditions with respect to ordinary matter and effective energy-momentum tensor for physically acceptable of wormhole solutions.