I'll pirate the Heroku docs later on how to set up the staging server, but the part that really threw me for a loop was getting a copy of my database on my staging server. This took many hours on StackOverflow before I finally messaged Heroku support for a ridiculously simple sounding answer.
heroku pg:copy app-being-copied::DATABASE DATABASE --app app-being-copied-to
Seriously. The word DATABASE. Not the complex database url, not any one of the fancy things I'd tried. Not the color of the database. The darn word DATABASE. Argh. There are hours of my life I'll never get back.