Unfortunately, getting one isn't all that intuitive. Let me break down easy for you. This will be focused on Austin, TX.
1) Get your birth certificate. If you don't have it, you can easily get a copy mailed by filling out a request and paying online
2) Make a copy of your drivers license, front and back. The post office will probably do this for you if you forget, for 25 cents
3) Some post offices will take your picture for you, but it'll be easier to just go to Walgreens. They'll take care of everything for you and it'll cost about 13 bucks. You'll get 2 photos. Don't smile in the photos, as no one is ever smiling when having their ID checked at the airport or customs.
4) Fill out the "online application" at https://pptform.state.gov/ This won't actually do anything but fill in the form for you and present you with a PDF.
5) Print the PDF (doesn't matter if its color or black and white) Make very sure your information matches your birth certificate, as this will be checked upon application. Do not sign it right now
Up to this point, the process is fairly well documented, but after this there's not as much info online.
6) You have all your stuff, but now you need to turn it in. You need to find a post office that accepts passport applications, Use https://www.usps.com/international/passports.htm to find a list.
Most of them will say they accept by appointment only. I called a few of these places and they all told me they were booked for at least a month.
There are a few that will list hours. These accept walk-ins. I had horrible luck with Bluebonnet station, and the google reviews seem to indicate everyone else did as well. I went to the Northeast Austin branch (on Blackson Ave) and there was no line on a Monday morning.
7) Go to the post office of your choice, go into the main area, and look for a signpost guiding you where to stand. At the Northeast Austin branch, it was furthest away from the entrance in a separate line from all the other business. Your application will be handled by one of the normal postal workers, so don't worry about trying to find someone "special" for passports.
8) One of the nice post office people will call you forward. They will take your birth certificate, the copy of your drivers license, one of your passport photos, and your application. They will also need to see your drivers license, so have that out. They will return your drivers license but keep everything else.
9) They will ask you to raise your right hand and affirm you're telling the truth about your application. Do so and answer the questions.
10) They will collect payment. This bit is tricky. Part of the fee is for the post office, and they can take any form of payment, but the passport part of the fee requires a check or money order to be sent along with your application. You can use a debit card or cash to create a money order, but NO CREDIT CARDS WILL WORK FOR THIS PART.
11) They will tell you when to sign your application
11) You're done Now you wait.
