  We present a series of algorithms for computing geometric and
representation-theoretic invariants of Calogero-Moser spaces and rational
Cherednik algebras associated to 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
$\mathbb{Q}$-factorial terminalization (and thus the number of non-isomorphic
relative minimal models) of the associated symplectic singularity.
