You have to research traits separately for every item type you can craft, but you can only research one thing at once PER CRAFT. But if you're doing more than one craft, say woodworking for staves and cloth for armor, you can have research going on both a staff type and an armor type at the same time.
IMO it's best to save your research time for the items you're actually going to use on that char; for instance, if you're a cloth wearer, research traits on cloth armor but not leather. One reason for this is that there are crafting stations in the world where you can make armor sets with special properties; but you need to have researched two traits (any traits will do) for each item you want to create in that set, and you need three set items to get the effect. So you might in fact want to strategize to make sure you get two traits on each of three items before branching out.
To ensure you always have something to research that is actually useful to you, make sure to save the items you find of the types you use that have traits you don't know yet; DON'T break them down, extraction is a different process from research and if you extract them you can't research them.
If you get a good item with desirable traits and want to wear it, that's fine; just remember when you replace it to research it rather than extracting it or vendoring it.