We present the Cosmological Double Radio Active Galactic Nuclei (CosmoDRAGoN) project: a large suite of simulated AGN jets in cosmological environments. These environments sample the intra-cluster media of galaxy clusters that form in cosmological smooth particle hydrodynamics (SPH) simulations, which we then use as inputs for grid-based hydrodynamic simulations of radio jets. Initially conical jets are injected with a range of jet powers, speeds (both relativistic and non-relativistic), and opening angles; we follow their collimation and propagation on scales of tens to hundreds of kiloparsecs, and calculate spatially resolved synthetic radio spectra in post-processing. In this paper, we present a technical overview of the project, and key early science results from six representative simulations which produce radio sources with both core- (Fanaroff-Riley Type I) and edge-brightened (Fanaroff-Riley Type II) radio morphologies. Our simulations highlight the importance of accurate representation of both jets and environments for radio morphology, radio spectra, and feedback the jets provide to their surroundings.