I visited the Trinity site in October 1997 (wow, has it really been that long?!? )
The site is on the White Sands missile range and only open the first Saturday in April and October. The site itself isn't very large and there's really not much to see. The site is very remote (not surprisingly!) They have a marker at ground Zero and a model of the FatMan bomb that was tested here and was also dropped on Japan. The ground all around has been scraped up and replaced but they have one small section preserved that has glass fragments created by the explosion. They had a display showing the radiation exposure for an hour visit is much less than an X-ray or even a cross-country flight. But still wouldn't hang around too long There's also an abandoned house a few miles away that you can visit. The entrance is near the town of Soccoro, New Mexico, off of US380.

I made a long weekend trip of it, I flew into El Paso Friday night and stayed overnight in Socorro. The next day I drove to the Trinity site, then continued onto Roswell, then finally overnight at Carlsbad caverns (loong drive). On Sunday I visited Carlsbad caverns then drove back to El Paso.