We present a series of algorithms for computing geometric and representationtheoretic invariants of Calogero-Moser spaces and rational Cherednik algebras associated with complex reflection groups. Especially, we are concerned with Calogero-Moser families (which correspond to the ${\mathbb{C}}^\times$-fixed points of the Calogero-Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig's constructible characters based on a Galois covering of the Calogero-Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package (CHAMP) by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a Q-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity.