On Windows simply download and run the installer (.msi). A shortcut to FamiStudio and the included demo songs will be added to your Start Menu. You will need install the VS2015 runtime (x86) which you can download here.
On macOS, download and uncompress the file anywhere and the launch FamiStudio app. The demo songs and included in the zip. The app is unsigned and might require you to ignore a few warning and you may need to grant it a few permissions if you are running Catalina.
A few demo songs are included in the installation folder (typically C:\Program Files (x86)\FamiStudio):
After The Rain (by Shiru, included with FamiTone2)
Danger Streets (by Shiru, included with FamiTone2)
Mega Man 2 - Stage Select (my approximate recreation)
Journey To Silius - Intro (my approximate recreation)