I am seeking a good introductory reference that could lead to an understanding of
the properties of geodesics in
complete CAT(0) metric spaces.
I am especially interested in learning the differences between geodesics in these spaces
and those in an $n$dimensional Euclidean space with its usual Euclidean metric,
which is of course CAT(0).
I am ultimately interested in simplicial, cubical, and polyhedral complexes,
but I am willing to start anywhere. Thanks for educating me!
_{[Image from "Shortest path problem in rectangular complexes of global nonpositive curvature" (Elsevier link)]}
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Your pictures remind me of the space of phylogenetric trees (which is cubical and exhibits similar folding), so if you are interested in reading about very concrete such examples, you might take a look at:
(1) proves uniqueness of geodesics in tree space, (2) gives a polynomial time algorithm to find geodesic distances, and (3) works in more generality. 


The canonical reference is Bridson and Haefliger, Metric Spaces of nonpositive curvature 

