We prove several theorems on the geometry and topology of random walks and random forests, with analysis of the latter of these random systems often relying on analysis of the former and vice versa. The main models we consider are the static and dynamic random conductance models, the uniform spanning forest, the arboreal gas and countable Markov chains, and we will be interested in both the qualitative and quantitative behaviour of these systems over large scales. The quantitative properties of both the random system and its underlying medium are in this work and in general often encoded as a set of dimensions, or exponents, which govern how those properties scale asymptotically with distance or time. In addition to the analytical work above, we numerically investigate the relationships between the dimensions of fractal media and the random systems which sit upon them, and, in particular, provide evidence that universality should hold beyond the Euclidean setting. Material taken from a total of six papers is included. We also include an introduction explaining the background and context to these papers.