Depending on where you are, you might be able to have your regular doctor or shrink call and set up an intake appointment directly with the hospital. This is a nice option because you don't have to go through the ER and the process goes faster. If not, just walk into the nearest emergency room and tell them that you are suicidal. Don't try to get into the long version or finer details with them. Just say you're suicidal and leave it at that. The process from there may vary depending on the hospital.
Generally speaking, they will then transport you by ambulance to the psych hospital or take you to the psych wing of the hospital where you will sit down with a bunch of different people and they will take your vitals, get your insurance info, medications, do a brief physical exam, and have you talk to a psychiatrist about whether or not you need to be hospitalized.
If they think you should (they probably will) for a voluntary admission they will ask you to sign a form stating that you give them the right to hold you for three days after you give them notice that you want to leave. After the three days are up, they have to convince a judge you need to stay which they rarely try to do. This means that if you check in, you can't check out before three days are up unless a doctor OKs it. I've definitely seen people go home earlier than 72 hours. If you want to go home early, it helps if family members talk to the doctors and say they will keep an eye on you or if you're going to another residential facility.
At this point they would search your belongings for anything sharp/metal/glass/cords/any toiletry items that contain alcohol like mouthwash/etc. These aren't necessarily gone for good, but they will keep them locked up and you may need to be supervised to use them. Basically, they take anything that someone on the unit could hurt themselves with. They might take away your cell phone or anything that has a camera for privacy reasons, so write down important numbers somewhere. Then they will show you your room (you will probably have a roommate) and get you a towel and bed linens. If you don't have anything with you, they can probably provide you with the basics like toothbrush/toothpaste, pajamas, socks, shampoo, etc. You will probably be more comfortable if you pack a bag in advance. I would go ahead and pack one now with about 3 days worth of clothes, that way if you decide to check in you can grab it and go -- one less thing to worry about. They will probably look in on you every 15 minutes at first to make sure that you're still breathing and not trying to harm yourself.
Those are the basics of checking in. I can elaborate on what to bring, etc. or if there's any specific question you have...
Checking in for the first time is a terrifying idea, but I think you will find that it's not so scary once you get there.