So far as I can tell, the American grad school interview process doesn't vary much from school to school; however, in the interest of full disclosure, I must warn you that my grad school interview sample size is two. Nevertheless, I managed to detect a good number of similarities between the two schools through my keen skills of observation. Generally, grad school visit breakfasts typically consist of orange juice, coffee, fruit platters, bagels with cream cheese, and some powder sugar covered pastries with mystery fillings. This final item will almost certainly give you stomach cramps and, as such, is best to avoid.
After breakfast and a game of guessing who is a prof, who is a current grad student, and who is a prospective, there will be a round of faculty talks to introduce you to the available research and reveal the answers to the earlier game. It's easy to be self-conscious about taking notes, but writing down first impressions and comments about research approaches and attitudes can be helpful in making a decision later. Writing down questions will give you something to mention in your follow-up email beyond the standard "thank you for your consideration." Taking notes also helps you stay awake if the coffee hasn't kicked in yet and shows the admissions committee that you are interested in the research being done at their school.
Lunch is, once again, standard: sandwich platters, chips, cookies, salads with the stupid bitter purple leaves in them, more fruit, and more sitting around trying to talk to faculty members without giving away everything you're planning to say during the interview. After lunch begins a series of 30 minute long interviews. In the absence of people with stopwatches waiting in the halls to intervene and shepherd prospectives from room to room, these 30 minute long interviews will inevitably be longer than 30 minutes.
If you're like me, the very idea of interviews is terrifying. I'm talking can't-get-any-sleep-the-night-before-shaking-in-your-seat-and-almost-passing-out levels of terrifying. At least, I used to react to interviews that way, but then I realized grad school interviews could be, well, fun. During grad interviews you get to talk about cool science with interesting people who have spent their career in a field you care about. You have a chance to ask them questions and see how their research is done. If you don't get along, if you find you don't have matching mindsets, that's fine, that's good! Then you don't have to spend half a decade working for somebody you don't like. But if you are motivated by the same goals and are able to communicate in complementary ways then you may have the chance to work long term in an environment that suits you terrifically! Your doubts about grad school may even be eased as a result of finding somebody you could imagine as a potential adviser.
Of course, after the interviews there will be dinner at a nearby, slightly more impressive than usual restaurant. There will be alcoholic drinks and it will be awkward to drink them. Prospectives will talk about everything except how the day went and the current students will smile and be relatively glad that they know what they're doing for the next few years of their lives, even if it is grad school. After a few hours, you'll go home, wonder if you met any of your future classmates, and get somebody to restrain you from checking grad cafe for the next week until you receive your final decision.
If you can treat your interview as an opportunity to discuss new ideas with other scientists, as well as stock up on free food, visiting day suddenly seems a lot less intimidating. For the first time in my life, I think I came out of an interview feeling more positive and confident than I did going in. And that's a feeling to look forward to. Good luck!