The Trinity Site is where the world's first atomic bomb was detonated in 1945 in preparation to drop the two bombs on Japan that ended WWII. "Fat Man," the plutonium bomb that was dropped on Nagasaki, Japan was mirrored after the bomb in this test. The site is a National Historic Landmark and is on the National Register of Historic Places.
The Trinity Site sits on the White Sands Missile Range in Socorro county, New Mexico which is about 2 hours from Albuquerque. White Sands is an active missile range, and the Trinity Site is only open two days a year to the public on the first Saturday in April and October. Gates are open from 8:00AM-2:00PM and it is first come first serve. You do not need to make a reservation and it is free to visit. There are bathrooms and food on-site, and you must bring your ID as well as proof of insurance & registration for the car you're driving. Google maps will not give you the correct driving directions. The site is semi-accessible so please read up before you go. You can find more info from the U.S. Army here.
I was staying in Albuquerque and headed to the site with the intention of getting there first thing when they opened. If you decide to go, and look at the directions from the U.S. Army that I linked above, this will make sense and be helpful: I turned on to state route 525 at 7:35AM, drove 3 miles and came to a stop because of all of the people in front of me. I got through the main gate at 8:37AM and then it was 14 more miles to get to the site.
There is a caravan from Alamogordo that arrives at the site around 9:50AM-10:00AM and it had multiple charter buses along with individual cars. That's when the crowds really swelled. My recommendation is to get there as early as you can, not only to beat the caravan crowds, but also because I don't know how busy it gets as the day goes on, and I wouldn't want to take the risk of it being a madhouse when it might be the only time you ever visit.
As part of the visit you go to ground zero and see the commemorative obelisk (pictured below). The only downside to getting there first thing in the morning though is that the sun casts a shadow on the plaque on the monument.
The building pictured below is also at ground zero, and you used to be able to see through the windows into a portion of the crater that hadn't been filled in and whose floor was lined with trinitite, but you can't see it anymore. Trinitite is sand that's been turned to glass, and in this case green glass (the picture below was taken through a glass case since trinitite is still radioactive).
You can also take a school bus ride to the MacDonald house which is about 2 miles away from the Trinity Site. Inside the MacDonald house was a plutonium assembly room and it's where the plutonium core was assembled for the world's first atomic bomb detonation. The house was originally built in 1913, sustained some damage from the Trinity blast, and was restored in 1984 to look the same as it did in 1945. The area around the house also has the remains of an underground cistern, a water storage tank, a windmill, and corrals and holding pens for animals.
I left the parking lot at 11:00AM and got back to the gate at 11:30AM. There was still a line of cars waiting to get in, but it was shorter than when I arrived though. All in all, if you're interested in the Manhattan Project, it's worth checking out. I have to say I was a little disappointed that the crater was filled in. I knew in advance that it would be, but I felt like it took away from the impact and significance of the site.