We have demonstrated transport of 9Be+ions through a two-dimensional Paul-trap array that incorporates an X junction, while maintaining the ions near the motional ground state of the confining potential well. We expand on the first report of the experiment in Blakestad et al. [ Phys. Rev. Lett. 102 153002 (2009)], including a detailed discussion of how the transport potentials were calculated. Two main mechanisms that caused motional excitation during transport are explained, along with the methods used to mitigate such excitation. We reduced the motional excitation below the results in the above reference by a factor of approximately 50. The effect of a mu-metal shield on qubit coherence is also reported. Finally, we examined a method for exchanging energy between multiple motional modes on the few-quanta level, which could be useful for cooling motional modes without directly accessing the modes with lasers. These results establish how trapped ions can be transported in a large-scale quantum processor with high fidelity.