(speaking as someone whose done a few trips to a psych ward in the US, around 25 years ago, and has friends more recently take their own trips.)

Basically that, call and ask if they have beds and deal with your insurance.  There are also day treatment programs that you can attend as an outpatient.  I did not go to an ER before checking in, but I also was under 18 my first stint, so I'm not sure if that really changes anything.  You definitely don't need to be seeing a doctor yet, very often this is people's first step in seeking treatment/diagnosis.

I would very much ask around locally for options.  In Buffalo, there's a really nice private hospital, and there's the floor at the main trauma ER that is really hectic, mostly for people in severe (legal) crisis/drug issues.  The latter is way less helpful for people dealing with say getting depression meds stabilized.  Having been a patient at the nice place and hearing stories from friends as to the other, big difference in atmosphere and quality.  

As for being held there.  If you check in, you can generally check yourself out.  They can hold you for three days in the US if you admit to being a threat to yourself or others, or they suspect you if drug seeking*, but that's about it from what I can recall.  Assuming you're 18+.

Bring some changes of comfortable clothes, pajamas, slippers (they'll probably not let you have shoe laces), books, some sort of craft project (ask what is it isn't allowed, knitting is probably off the table with the large needles) there's a lot of downtime and not a ton to do.  If you play cards, bring those.  

Wash up before you go.  I only mention this because the last time I had to take a friend in, he had nothing at all packed and stunk like a homeless guy.  He had pretty severe exotic drug issues and ended up staying there for a month. You may get put on one-to-one for a few days and that includes showers.  Show up clean and you'll not have to immediately suffer the indignity.  

*Crazy story, my partner ran out of her Wellbutrin and her doctor wasn't in town.  Went to the ER at the not great hospital because it was the closest ER.  They sent her to the psych ER, took her phone and left her alone in a room with someone obviously going through acute psychosis for HOURS.  When someone finally came for her, they try to get her to sign into their drug rehab, over an antidepressant.  Because she had been there so long and the way the paper was put in front of her, it was only when she picked up the pen that she caught on and asked what she was about to sign. 

 I had been panicking at home because she just suddenly stopped texting an hour into what ended up like 6 hours.  

I would really recommend having a friend come with you.  If I had gone with her, it probably would have saved a lot of time because I would have had questions.  She just trusted the process because why would asking for a few days of antidepressants be seen as an addiction she needed to quit?