I think New York is probably the greatest city in the world, but the most fun to visit has to be Rome. The beauty of the city combined with the churches, muesums, ruins, shopping and people can't be beat.
In Texas, my favorite city is probably Austin. (I love Houston, but I've lived in the area most of my life so I won't count it!) Outside of Texas, my favorite city is probably Boston. I spent several summers there as a teenager and it was such a neat place to hang out.