When asked to update user preferences, select "Yes".
Select fcitx from the list, and select "OK".
Restart your computer, and you should see an icon for fcitx in the wingpanel.
NOTE: For newer versions of elementary OS like Juno and Hera, you probably won't see the fcitx icon in the wingpanel at this point. These instructions fixed that issue for me under Hera: https://elementaryos.stackexchange.com/a/16503
Click on the icon and if Mozc is not already under the Input Methods, click on configure. Click on the Plus icon at the toolbar at the bottom. Uncheck the checkbox for "Only Show Current Language". Search Mozc, and add it.
Ctrl+Space is used to switch between inputs.
Since the mozc configuration tool will not open from the indicator, you can use the following command to open it. /usr/lib/mozc/mozc_tool --mode=config_dialog
If everything went well, you may have noticed that the icon for mozc by default is too big, so wingpanel will resize everytime you input with mozc. To avoid this, you can replace the icon with this smaller one in /usr/share/fcitx/mozc/icon/. You can download a copy of the mozc.png icon here, make sure you replace the old mozc icon with this. Once you replace it, open up terminal, and enter chmod 777 /usr/share/fcitx/mozc/icon/mozc.png, to ensure that the icon will be displayed properly.
You can change the keyboard layout in System Settings > Keyboard > Layout and add a new layout there with the plus button in the bottom left. I have no knowledge of Japanese signs, but enabling a Japanese (Kana) seemed to work for me.
The default shortcut to cycle through different keyboard layouts in Freya is AltSpace. You can change this in System Settings > Keyboard > Shortcuts > Universal Access.
Also there are some special settings for Japanese input(Keyboard -> Options -> Japanese keyboard options). Check attached screenshot: