WHAT TO DO IN COURT
Most people don't have to show up to court every day, but as an attorney it is a regular part of business, and it is always something that I talk to my clients about before we get in there.
WHAT TO WEAR
Whether you are there for a speeding ticket, a felony, or a plaintiff trying to get money to cover your medical expenses after a car accident, make sure you dress the part. While attorneys are expected to wear suits (for men) or business attire such as pant suits, skirt suits, or some other acceptable clothing (for women) the clients don't have to arrive in the same level of dress. But it won't hurt if you do.
I have been in several courts where the Defendant (this was criminal law) showed up wearing shorts and flip-flops and the Judge wouldn't even hear their case until they put on something more respectable. If you arrive wearing a suit, or a dress shirt and slacks the judge is going to treat you better. They may even listen to you when you ask for some special treatment. If you don't look like you care about being in court, the judge won't care about you.
HOW TO ACT
Maybe a better title is "How not to Act." If you are not sure what to do when you are in the courtroom, then just stay quiet, it almost never hurts to just quietly wait until you are called. Don't walk up to the podium or past the desks in the court room until you have been called to. Call the judge "Your Honor," they like it. Don't chew gum when you come in, and if they let you have a phone in the courtroom (most places don't) do not play on it while you are waiting.
What the judge really wants to see (especially in a criminal case) is that you understand that this thing is serious. They want you to treat the experience like it matters. A good rule of thumb is to ask what would do if your grandmother was watching you in church. Don't swear, don't make a lot of noise, just be patient. If you need to talk with someone, especially your attorney, then step outside and have a quiet conversation where the judge can't see or hear you.
Lastly, get an attorney. Don't just go out and stop in to court without one unless you just can't afford it. You need to have someone there to take care of your interests. You need someone trying to do what they can to make your life better. It may cost a little more than you want up front, but you will save yourself time, stress, and potentially even money.
Good luck, and don't forget to call or send an email if you need some help.