A visit to the Supreme Court of the United States (SCOTUS) involves a lot of waiting. On a normal day—one where the cases the court is hearing aren't exceptionally well-publicized or controversial—you should plan to arrive by 8 a.m. at the very latest. Be sure to get a number from the guards near the front of the line, then wait. The waiting is exceptionally fun because the sun rises behind the Court building, ensuring you'll stand in the shade for at least part of the time. Think "cold." At around 9:15 or 9:30, the guards will let the first 50 people into the Court. If there's space (i.e., if a group w/reservations doesn't bring the numbers it reserved space for), the guards might let another handful of people in from the line, but it didn't happen the day I was there. I had number 59, so I didn't make the first cut. I arrived at 8 a.m.; if you really want to make sure you get in, arrive earlier.
Arguments begin at 10 a.m. and again at 11 a.m., so if you miss the first cut, you can wait and hope some people leave after the first argument so you can get in for the second. That worked for me; they let 25 people in at about 10:40. It felt great to get inside out of the cold! Plus, the building itself is beautiful (at right is the ceiling in the entryway), so there's plenty to entertain once you make it in. We waited until 11:10 or so, when we were finally ushered into the little chairs at the very back of the courtroom to watch the Justices and attorneys do their thing. Very cool. Highly recommended if you visit D.C.