Here's of those things that are on the surface straight-forward but in practice require a few steps that might not be easy to remember for future (re-)installs. Here are the steps needed to get Korean input working under Ubuntu 19.10 (and presumably 19.04, too).
Installing required components
First, head over to your system's Settings and go to the Region & Language section. Click the + button under the list of Input Sources and find and select Korean, then select Korean (Hangul) and click the green Add button.
Next, click the Manage Installed Languages button. This window that pops up might mention that not all tools/bits are fully installed yet, so click yes to have it automatically install whatever bit was missing. Next, click Install / Remove Languages.., find Korean, enable its checkbox and click Apply. It might already be enabled if when you added the Korean input option, but it's good to check here to make sure.
In this same window make sure that Keyboard input method system is set to IBus.
If you actually want your entire operating system to be in a different language, you would install this language in this window and drag it all the way to the top of the list. Because I prefer my operating systems in English, I did not do that.
Note: This isn't entirely obvious in the UI here, but know that all language options listed under English are grayed out on purpose. Only languages placed above English actually show up as enabled. I thought it'd mention it here as it wasn't entirely obvious to me what was happening here at first.
Next, open up the Ubuntu Software app and search for
ibus-hangul. If it's not already listed as being installed, install it now.
At this point you should log out and back in again. Or if you have been holding off updating your system anyway, run apt update & upgrade and go for a full reboot to ensure your entire system has everything fully loaded and is up-to-date.
Configure the input type
After doing this, you should now be able to write Hangul at long last. The system-wide
super+space keyboard shortcut switches between the different input sources, and when in the Korean input method you can use
shift+space or your keyboard's dedicated hangul toggle button if it has it to switch to and from Hangul mode.
It seems like there is currently a bug with Ubuntu 19.10, IBus or Hangul input specifically that prevents the icon from showing up in the system tray. I don't know if this is specific to my setup or if something else is going on, but for the time being I avoid this issue by relying on the system default input switch method, and basically avoid using the IBus specific one.
Click the settings cog next to Korean (Hangul) in the Input types list and enable the Start in Hangul mode option. This way whenever you switch to Korean it will immediately start in Hangul mode instead of its own english/roman input mode.
I personally kept running into accidentally switching to/from Hangul because of that default
shift+space shortcut, so I removed that one too as you can see above.
As I mentioned at the beginning, it's not very complicated to set this up, but due to some odd choices and naming conventions it can pretty confusing. Simple things like having both a Korean input option (which won't let you write Hangul) and a Korean (Hangul) specific input option that does let you write Hangul are kind of confusing. I guess the non-Hangul supporting Korean option is just for a Korean-specific keyboard layout, not actual input type? Who knows.
Regardless, I hope this may help you set up the ability to write Hangul on your Ubuntu setup.
Thanks for reading!