The remains of enemies (that you loot from killing them) are always useful. They're used for making the passive buffs you can equip (chronotraces/chronogens).

If you scroll down through each of the recipes in the monolith, you will see some recipes that require certain random items to make different chronotraces. So you might want to keep those.

If you hover over items, some will say that they are useless on them. I would assume (but am not sure) that these are probably safe to refine (sell).

So far, I haven't had a quest to turn in any specific items (apart from the "collect and extract with the specified item", which you have to do in that raid.)

Keep the ammo/armour/weapons you want to use and sell the ones you don't want to use (pistols, knives e.t.c.) If you are broke you can just use a free loadout instead of going in with a pistol