If you gather resources or pick up items, they currently will be placed in the first free slot of your inventory (from left to right). Because of this, I'm constantly sorting my new items from the most-left slot into various bags I'm carrying.
My idea: I think bag behavior would be more convenient and logical if items your gather/pick up will go to the first free slot of the currently active/opened bag.
This way I can simply open my 'Mining' bag if I go mining, and my rocks will directly go into the right bag. And if I'm tending my garden, I open my 'Gardening' bag. Easy.
* I know there are containers (like toolboxes and spice-cases) that sort of do this already, but for lots of items these specific containers do not exist.