To download and install GameMaker Studio 2, you must first create and sign in to a YoYo Account (either an account you have made previously as an existing user or a new account if you are new to GameMaker), and then on the Licences page you will see a Download link.
Clicking this will initialise the download of the GameMaker Studio 2 installer, which you should then execute when the download has finished.
Installing On Windows
When you first start the Windows installer package, you will be prompted to agree to the YoYo Games Licence Agreement.
IMPORTANT! If you are updating an existing version of GameMaker Studio 2, you will be asked if you want to first uninstall the previous version before being prompted to accept the licence agreement for the new version.
Once you agree to the licence, you will then be taken to the Components screen where you can change certain options, like whether to add a desktop shortcut or link yy* files with GameMaker Studio 2 (note that some options are obligatory and can't be un-checked):
Clicking "Next" will take you to the install path for GameMaker Studio 2. You can leave this as the default path - which will usually be your C drive - or you can add a custom path for the program to install to:
IMPORTANT! While GameMaker Studio will install to the path you give, certain components will also be installed in the %programdata% folder, as well as the %localappdata% and %appdata% folders. You can change these paths after GameMaker Studio has installed, from File > Preferences
Clicking the "Install" button will now install the program, showing the installation progress as it goes. Once installed, you can click the "Finish" button and the installer will close and GameMaker Studio 2 will start. Note that when GameMaker Studio starts, there me be additional downloads required by the program, but these are all handled by the IDE.