Question: I would like to be able to change the output device sometimes, for example to switch between headphones and speakers, without having to use the mouse.

I've figured out how to focus the 'status menu' area (top-right of screen) with a keyboard shortcut (default is `Ctrl`+`Fn`+`F8`), and can select the sound menu.

But when I try to select items in that menu, pressing the arrow keys will only alter the volume (even with modifiers like `Cmd` or `Alt`). Even `page up`/`page down`!

How can I access the rest of the menu with the keyboard and change the output device?

Answer: 
I couldn't find a way to do this using the status menu, but another quick way by keyboard is:

1. Open Sound Preferences (`Opt`+`Volume`  -- any of mute/volume up/down will work)
2. Use `tab` + arrow keys to navigate to the "Output" tab if necesssary
3. `tab` into the menu and use arrow keys to instantly change output
4. `Cmd`+`W` to close the window

(macOS 10.15)
The first answer to [Using Apple Script to Manage Sound Output Selection](https://apple.stackexchange.com/questions/217148/using-apple-script-to-manage-sound-output-selection) describes how to use an AppleScript to swap between two output devices. This works for me without modification.

There is also [Applescript: "can't get tab group 1 of window"](https://apple.stackexchange.com/questions/209352/applescript-cant-get-tab-group-1-of-window-el-capitan) which sets to specific devices. It is rather old, so may need some debugging.

Once you have a working script, there are various ways of running it with a keyboard shortcut. The simplest is to use [FastScripts](https://help.redsweater.com/fastscripts).