I thought the new destroyers were also nuclear.
DDG-51 destroyers are Arleigh Burke class. There's 55 of them so far; none are nuclear powered.
Rep. Gene Taylor made some noise about canceling the Zumwalt/DDG-1000 class (gas turbine-powered) in favor of a nuclear Burke variant, but it hasn't happened.
Wasn't there a big show of the all nuclear carrier group that could go around the world with having to refuel?
What you are referring to is 1964's Operation Sea Orbit. You need more than a carrier and two missile cruisers to make up a carrier strike group. Nuclear-powered destroyers and supply ships were not built, and the all-nuclear Navy never materialized.
Since the last nuclear cruiser was decommissioned in 1999, the only nuclear vessels in the US Navy are aircraft carriers and submarines.