Items

There are 3 item types: 

•  Training
• As the name says, these ones are used in order to improve your experience into training, you may have INT, TEQ, STR, AGL and all type training items.
• How to get it: Surfing through the maps, as well as the daily training event, from 19:00 to 20:00 PST. 
• Item color: Inner color is "RED", outside color is White (Common), Bronze (Rare), Dark Silver (S. Rare), Gold (U. Rare)

 

  Awakening Medals

• These are used in order to upgrade your hero (using the awakening option), you may have INT, TEQ, STR, AGL and all type items.
• How to get it: Surfing through the maps, as well as the daily Awakening event, from MON to FRI. (Blue, Green, Purple, Red and Yellow). There is also another events which can give you particular Awakening medals, like "A lone Warrior's Last Battle", which give the needed medals to Z-Awaken Bardock-R to a SR.
• Item color: Inner color is "Pink" or "Purple", as you prefer, outside color is White (Common), Bronze (Rare), Dark Silver (S. Rare), Gold (U. Rare), which with a very LOW chance can be a, let's say Ultra Ultra Rare, for SSR Characters.

•  Support

• These are used in battle, to remove status, heal, increase attack, decrease enemy's attack and so on. (you can read the description clicking on it.
• How to get it: Surfing through the maps, there is no support Event yet.
• Item color: Inner color is "Blue", outside color is White (Common), Bronze (Rare), Dark Silver (S. Rare), Gold (U. Rare).

Items can be common, Rare, Super rare, Ultra Rare and, some other name that I'm unaware (never got one of these, often called as Ultra Rare, however can be named as Ultra Ultra Rare), something near to impossible, other ones are given by accomplishment of Events.

Here are the Full database of items (Japanese site translated by Google): 
gamy.jp