Advice From Previous Graduate Students

What to wear, outline of process, formalities.

For attire, you do not need to be overly formal (don't wear a suit). Do not be too casual either, dress nicely like you would to teach a class.
Be a few minutes early. After you are done with the exam (board presentation or paper presentation), the committee will immediately deliberate while you wait in the hall. You will find out within moments whether or not you passed.
They will ask a question you do not know. That is part of the process. It is okay to say you do not know or ask for the question to be rephrased.

(Analysis Student) Make a solid presentation so that questions will be minimal. If no stones are left unturned, you can’t be stumped during your exam.
(Topology Student) Work through as many questions as possible. Focus on understanding the techniques rather than just understanding the solutions to specific problems. Understanding the techniques helps you widen the way in which you approach different problems so even if something that you didn't prepare comes up during the exam you could trust your intuition and try and solve it there and then.

You are going to mess up often. That's okay. It's part of the learning process. Just keep working hard!
Your committee members would always try to give hints and direct you into solving the solution, so be confident. It is indeed stressful but you got this!
It is okay to have "wacky" ideas, try to share them with your advisor every now and then.
Meet with your advisor every week if possible; it helps you stay on track and have a meeting to work towards.

If you have an idea about the area that you will be working in, get an account on arxiv as soon as possible. You can then subscribe to areas of your interest and receive emails whenever new work in those areas gets published on Arxiv.
Talk and have discussions with other people in your field, not just your advisor.
Do not be afraid to email authors of papers you find interesting. They often are excited to hear you read their work!