We use contact homology to distinguish contact structures on various manifolds. We are primarily interested in contact manifolds which admit an action of Reeb type of a compact Lie group. In such situations it is well known that the contact manifold is then a circle orbi-bundle over a symplectic orbifold. With some extra conditions we are able to compute an invariant, cylindrical contact homology, of the contact structure in terms of some orbifold data, and the first Chern class of the tangent bundle of the base space. When these manifolds are obtained by contact reduction, then the grading of contact homology is given in terms of the weights of the moment map. In many cases, we are able to show that certain distinct toric contact structures are also non-contactomorphic. We also use some more general invariants by imposing extra constraints on moduli spaces of holomorphic curves to distinguish other manifolds in dimension $5.
