Install Tweaks by running:

sudo apt install gnome-tweaks

Then go to Tweaks → Fonts → Scaling Factor and change the scaling value.

At least this option helps to make text bigger.

Actually I like this approach, since it keeps other things at scaling 1, because I only need the text to be bigger.

XOrg solution

The Archwiki proposes a solution (or rather a hack) with xrandr (if you use standard Ubuntu 18.04 with xorg):

ArchWiki - HiDPI - fractional scaling

Fractional Scaling

A setting of 2, 3, etc., which is all you can do with scaling-factor, may not be ideal for certain HiDPI displays and smaller screens (e.g. small tablets).

xorg

You can achieve any non-integer scale factor by using a combination of GNOME's scaling-factor and xrandr. This combination keeps the TTF fonts properly scaled so that they do not become blurry if using xrandr alone. You specify zoom-in factor with gsettings and zoom-out factor with xrandr.

First scale GNOME up to the minimum size which is too big. Usually "2" is already too big, otherwise try "3" etc. Then start scaling down by setting zoom-out factor with xrandr. First get the relevant output name, the examples below use eDP1. Start e.g. with zoom-out 1.25 times. If the UI is still too big, increase the scale factor; if it's too small decrease the scale factor.

xrandr --output eDP1 --scale 1.25x1.25

Wayland solution

Since Ubuntu 18.04, Wayland is the default display protocol.
To enable scaling:

• Enable fractional Scaling experimental-feature:

   gsettings set org.gnome.mutter experimental-features "['scale-monitor-framebuffer']"
  

• Restart the computer.

• Open Settings -> Devices -> Displays

• Now you should see 25 % step scales, like 125 %, 150 %, 175 %. Click on one of them and see if it works.

I run a shell script on login (using Startup Applications control panel) to set my desired scaling.

Determine your output device (mine is DP-1) by running xrandr on its own.

Then put an executable script somewhere (I have it in my home directory) containing the following:

#!/bin/bash

gsettings set org.gnome.desktop.interface scaling-factor 2
gsettings set org.gnome.settings-daemon.plugins.xsettings overrides "{'Gdk/WindowScalingFactor': <2>}"
xrandr --output DP-1 --scale 1.5x1.5
xrandr --output DP-1 --panning 3840x2160

Tried to play with expermential features and some other "tips" found online but nothing helped or seemed like "solution" will introduce another pallet of issues. What i eventually did is - i installed unity desktop manager which was used with Ubuntu 16. So now I have running latest 18 LTS with a bit different desktop manager which does have fractional scaling without any hacks.

sudo apt install ubuntu-unity-desktop

I used Unity Tweak Tool in Ubuntu 16.04. I'm running now a test environment where 16.04 was upgraded to Ubuntu 18.04 LTS. I just checked and Unity Tweak Tool is still there and scaling my High DPI screens properly. The settings I'm using for scaling on a 1920x1080 monitor are:

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

Icon size is preset but the text size under it increases giving the illusion the icon size changed:

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.

To install Unity Tweak Tool use:

sudo apt install unity-tweak-tool

Others may be interested in the full suite of tools available 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 can't, because Ubuntu switched back to Xorg as the default display server. A lot of people are searching for a solution, and this thread is the top result on Google. Unfortunately there is still no fractional scaling in Xorg.

You can use 1600x900 (16:9) from dropdown to increase scaling.

I am currently experimenting with a fix for this (still using Ubuntu 16.04 admittedly, but I'm pretty sure the same options are available in 18.04) which keeps the scaling at 1, but uses the resolution selector in the "Screen Display" part of System Settings to drop the resolution of the panel. You would think this would make everything blurry, but it seems to be doing a fine job for me. YMMV.

My screen (X1 Carbon 4th Gen) is 2560x1440, a 16:9 ratio. 2x zoom would make it effectively 1280x720, which is too small. Instead, I pick 2048x1152 (also a 16:9 ratio) from the selector. For me at least, it also offers 1920x1080, 1600x900 and 1368x768 as options depending on how much downscaling you want.

I might end up going with 1920x1080 as 2048x1152 still has a few things a little bit small.

The solution with the Gnome Tweak Tool works well for me with Ubuntu 18.04 with a 3840 x 2160 resolution display. After setting only the scale factor to 1.5 and leaving the font sizes unchanged the screen display was great for me. Additionally I had to adjust the size of the dock symbol size in the gnome settings. That seems to me the least intrusive solution.

The 18.04 desktop is fitting wallpaper images according to width, so if you resize your images for the width of the screen (regardless the absolute size of the image) your wallpapers will fit. To fit the tall images I go into "alter image/change canvass size" and add wide borders on either side using Pinta. (or Gimp or whatever you've got)