A great place to start is Peter Scott's beautiful paper "The geometries of 3-manifolds".

It goes through tons of examples of 1- and 2-dimensional orbifolds; this includes those coming from spherical/Euclidean/hyperbolic triangle groups, and he gives a complete classification of 2-dimensional Euclidean and spherical orbifolds. Along the way he covers all the usual tools like orbifold fundamental group, van Kampen's theorem, and Euler characteristic. Finally he uses orbifolds heavily, in the context of Seifert fibered manifolds, to explain which manifolds admit one of Thurston's eight geometries, and how. The other topics touched upon in the paper (Lie groups, connections and holonomy, group actions, foliations, group extensions, etc.) are all well worth knowing, and presented very explicitly and clearly here.

(A scan is available from Scott's webpage, but it is oriented sideways and the filesize is large. Anyone with access should get the paper from the journal directly.)