How to save/export all the settings of IntelliJ IDEA?

File --> Export Settings...

AND don't forget to backup your Run Configurations!
Go to where your workspace is located, there is a .idea folder. (i.e: /Users/matthew/codebase/.idea)

Copy the runConfigurations folder and paste in your new workspace. That's it!

All your settings are in a configuration directory that depends on you OS. You can just take the entire directory. Mine is in my home directory C:\Users\Me\.IntelliJIdea12 Just placing that in the same location on your new os before you install IntelliJ should work. It is what I do myself.

For the new IntelliJ IDEA version 2020.03 version:
File → Manage IDE Settings → Export Settings...

Sure : File → Export Settings.

2020.01 version

This is now out-of-date. You have to go File --> Manage IDE Settings --> Export settings. You can disable the "Settings repository" under File --> Settings --> Plugins but this doesn't seem to make a blind bit of difference to your ability to export settings "manually".

In the later versions it seems like you should go with either configuring a settings repository or using IDE Settings Sync plugin. More information here:

IntelliJ IDEA lets you share your settings between different instances of your IDE. This helps you recreate a comfy working environment if you are working from different computers and spare the annoyance of things looking or behaving differently from what you are used to.

You can share your settings in one of the following ways:

By configuring a settings repository. This allows you to sync any configurable components (except for the list of enabled and disabled plugins), but requires setting up a Git repository with the settings you want to share. This option is useful if you want to implement the same settings among your team-members.

By using the IDE Settings Sync plugin. It utilizes the JetBrains server, so no additional configuration is required. Synced settings are linked to your JetBrains Account, so they will not be available to other users. The settings you can sync include: IDE themes, keymaps, color schemes, system settings, UI settings, menus and toolbars settings, project view settings, editor settings, code completion settings, parameter name hints, live templates, code styles, and the list of enabled and disabled plugins.