County courthouse? I understand that there are different places, but I thought that post offices were the most common. Then there are city or county agencies. We did our kid's at the UC Berkeley rec gym.
http://recsports.berkeley.edu/about/member-services/get-a-u-s-passport/
Really though, the best way to do it would be online. That produces a PDF file that you can print with all your entered info as well as a bar code. The State Department will have all that information available on their servers, and all they need to do is scan the bar code to bring up your file. That means less chance that someone on their end makes a typo or gets basic info incorrect. You can also make corrections if you make a mistake since it's not official until you send in your application on paper. I've done it that way for my last renewal, as well as for our kid's and my wife's first-time applications. When we were applying for our child, someone in front of us was taking nearly forever filling out the form, which he got at the facility.
If you are eligible to mail it in, you're not going to need to pay the $25 fee to the "acceptance facility". You'll need to pay for mailing costs though, which you wouldn't in person. The other pain is that if you write a check, you'll need to make out a separate one for the facility (or pay cash), as well as one for the State Department.