That is, does NASA still have all the training equipment, computers and so forth that would be required for a new moon program, or would all of that have to be rebuilt as well as building new rockets?
No Apollo training equipment is still functional, at least at Johnson Space Center (JSC).
The Apollo Mission Simulators were removed and sent to museums. The facility space they once occupied in Building 5 was then filled with the Shuttle Mission Simulators. They have now been removed and sent to museums.
The sole remaining Lunar Landing Training Vehicle hangs from the ceiling in the lobby of JSC's Teague Auditorium. It is in a safed museum configuration. It is not inconceivable that it could be returned to flight status but it is unlikely.
The Skylab simulator was removed from Building 5 and placed on display in Space Center Houston.
First, just about none of the original equipment is in usable condition. It may not be completely impossible to refurbish it, but it would be so far beyond the realm of the achievable that is might as well be. If you really had your heart set on it, some Saturn V's were built for Apollo 18-20 but never flown. Not sure about LEM and capsules though. Second, were all that in usable condition, there's no way that it would be politically viable to make people's lives depend, very visibly, on technology 40 years obsolete.