In the era of precision cosmology, a wide range of cosmological surveys, such as the LSST, DESI, Euclid and WFIRST will precisely probe the large-scale structure of the universe, shedding light on the nature of the dark sectors. Given how sensitively the growth of structure depends on the nature of the underlying gravitational field, this will be a unique opportunity to constrain the so-called Modified Gravity models (MG), that are theoretical alternatives to dark energy, which attempt to explain cosmic acceleration through a large-scale modification to General Relativity. In order to fully utilize the wealth of incoming data, however, theoretical predictions of structure formation in such alternative scenarios are necessary. Due to the existence of an additional degree of freedom that these models introduce, N-body simulations prove to be highly computationally expensive. In the first chapter of this thesis, we discuss how we can overcome this issue by using Lagrangian hybrid techniques, which can lead to a speed-up by 2 orders of magnitude, compared to the conventional tools, while still achieving % level of accuracy. Then, in chapter 2 we proceed to introduce novel statistics that can help us more confidently detect MG signals hidden in cosmic density fields, by up-weighting the significance of cosmic voids, where the MG-LambdaCDM degeneracy is broken. In the scales where structure formation is analytically tractable, finally, we show that we can make accurate analytical predictions for the two-point statistics of halos in MG, using Lagrangian perturbation theory and the Gaussian Streaming Model, simultaneously capturing, for the first time in modified gravity, the effects of both halo-bias (in chapter 3) and redshift space distortions (in chapter 4), effects crucial for the interpretation of photometric and spectroscopic observations. Our results demonstrate that a series of analytical, semi-analytical and simulation-based tools can be utilized in order to dramatically improve our understanding of the nature of cosmic acceleration and gravity at cosmic scales.