How do I increase the font size in Ubuntu 18.04 LTS for a HDPI / 4K display?

You may use (GNOME) Tweaks to adjust font sizes of various elements. First install Tweaks by running

sudo apt install gnome-tweaks

(or sudo apt install gnome-tweak-tool).

Then launch Tweaks and go to the Fonts section, you'll get the options to change font sizes.

You may use (UNITY) Tweaks to adjust font size along with other UI elements. Install it using:

sudo apt install unity-tweak-tool

The settings I'm using for scaling on a 1920x1080 monitor are 1.38 times. You would probably want 2.00 or higher:

The Text scaling factor scales both fonts and UI elements like title bars, menus, etc.

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Here's a GIF showing changing scaling from 1.38 to 1.00 and then to 2.00:

In the .gif above scaling starts at 1.38 on a 1920x1080 monitor. Then it is changed to 1 and everything gets tiny, which is normal. Then it is changed to 2 which is ideal for the visually challenged. Once again the icons have fixed pixel size and the font shrinking or expanding under the icon gives the illusion their size is changing.

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There are others tweak tools that may be of interest in 18.04 LTS:

$ apt list | grep tweak
gajim-rostertweaks/bionic,bionic 1.0.0-3 all
gnome-tweak-tool/bionic,bionic 3.28.1-1 all
gnome-tweaks/bionic,bionic 3.28.1-1 all
mate-tweak/bionic,bionic 18.04.16-1 all
mousetweaks/bionic,bionic,now 3.12.0-4 amd64 [installed]
tweak/bionic 3.02-2 amd64
unity-tweak-tool/bionic,bionic,now 0.0.7ubuntu4 all [installed]

You don't need to install third-party tools if you using Unity.

Just go: Unity control center -> Displays -> Scale for menu and title bars

Now you can scale everything to a readable size: