Thanks ... Looks like I'm making progress, because I now at least hear the system notifications.
You have Pulse Audio installed ....... is it enabled and working correctly?
Do you have pavucontrol installed from the repository?
If not, install it, and launch PulseAudio Volume Control.
If you get an error please report the error in full.
If no error, you can turn off the HDMI sound card on the config tab which will force the use of the onboard sound.
If you wish to use the HDMI then on the playback tab select the output device for each application you launch to pay sound.
I installed pavucontrol, which I think also installed PulseAudio as a dependency. Somehow I thought that PulseAudio was already installed.
This action seems to have enabled the system notifications, but not other audio (e.g. audio from the browser).
But when I click on PulseAudio Volume Control in the Sound menu, I get 'Connection failed: Connection refused.' Do I need to run this as root? If so, what's the command-line way to run this?
I should have mentioned this, but my speakers are integrated into my monitor, which is hooked up by HDMI. I don't really care if I use HDMI sound or the onboard sound, I'm just looking for the simplest solution here.