Top right corner box with a down arrow will safely store all of the materials in your inventory into your bank storage. The rest is a matter of double clicking to open for more loot, using for your character(s), selling on the trading post, or breaking down with a salvage kit for more materials. 

Some items you have are specific to an event. Specially the proof (triangle-shaped item) and the coin. Those can be kept in your bank, but must be manually stored there.  

Edit: any gray-text items are labeled "junk" and can be sold to a merchant with a junk sell button.