The first test site in the US, in New Mexico within what is now the White Sands test site, can be visited by a casual visitor with no precautions or special clothing required. This test, Trinity, was a plutonium bomb, exploded in July, 1945.
Another test site in the Marshall Islands in the southwestern Pacific Ocean, Bikini and Eniwetok, are off-llimits to all due to the effects of five tests done there, both atomic and hydrogen bombs in surface, air, and subsurface explosions. There is vegetation and plant and animal life on the islands, but residual ground/water contamination makes the area uninhabitable.
In Nevada a desert test site in the southwest corner of the state site was chosen due to its being empty, barren, and isolated land. The site was used for over 900 nuclear tests (800 of which were air-bursts), and the area remains off-limits to the general public for several reasons, including its continued use as a nuclear waste disposal site. Several hundred govt employees do work there.