It wasn't uncomplicated, but we managed to go on a White House tour a few days ago. We did count it as one of the DC history field trips even though it's not quite right for the class. But the White House is obviously a major building in DC and since there's no way you can time your visit to match up with any certain time period since you have no control over when you get to go, and since it doesn't match any one time period any way, we just went when they said we could.
You have to contact one of your congresspeople first and they ask you for all kinds of personal information and the dates you are interested in going. Just the dates. You can't pick times which was irritating since I can go just about any day at 11 AM but I can never go at 8:30 AM. It's a lot different when you're making the request as a tourist who's visiting the city because you can plan your visit around the time you get instead of as a resident who has a school bus to deal with every morning and afternoon.
You can't take much of anything in with you, but you can take a camera now. The rule was recently changed and a lot of the info, including what you are emailed and the posted signs at the White House, will say cameras are forbidden, but at the last minute there's a sign that says they're fine. But really, there are better public domain photos that you can find online unless you are in need of a selfie. I skipped the camera and don't regret it.
Since you can't take anything with you, I ended up bringing a jacket just for the pockets since I hate to fill up the pockets of my jeans with stuff. We went home right after because we didn't have anything with us. Just the metro card, a credit card, a house key, and ID. You don't need to print any of the papers. Just show up.
Our time was scheduled for 11:00 and we got there a little after 10:30 and got in line then. No one cared what our exact time was (maybe they would if you were an hour early or something?). All the people around us in line were also scheduled for 11 AM. It took a while to get through all the lines and ID checks, so I think we got inside at about or a little after 11.
When you're inside, you can look around as you please. They allow you to go to a very limited area, obviously, but there's no rush. The Secret Service people know lots of things and you can always ask them questions, or just do what we did which was to ask them to tell us about the room we were in so I didn't have to think up a specific question.
It was interesting but not something I would have gotten around to without this class, and I didn't change my mind about that after going. It is inconvenient to schedule the tour and to fit it in your day so don't bother unless you really want to go. The new White House visitor center is much easier to visit and really interesting. That's a good option to do instead.