I'm getting a lot of low quality - awful, shoddy, etc items in my storage areas and I don't know how to get rid of them or destroy them. I could move them to another area, but that doesn't really solve the problem.
With low quality items you generally have three options.
Firstly, the method I recommend is smelting the item if it is a long range/close quarters weapon. You can use a smelting table to turn the weapon back into it's original material which can then be re-crafted into a new weapon or used for building something else. Electric Smelter in rimworld
You can sell it, you simply need to put down a trade beacon and wait for a trade to come by. Because the weapon is poor quality you most likely won't get much money for it but it's better than just sitting there. In the newest update you can get trade caravans that come more frequently, but they often won't accept items and have little wealth. Trade Beacon
You can move it to dumping pile/random location. With the dumping pile you can specify different levels of quality you want to filter for dumping. The same can be done with stock piles, so you can exclude gathering items that are of low quality. Another option is to have one of your colonists equip the item, take control of them, guide them to a random, far away point on the map and make them drop it, then set it to not gather.
Leave it outside in the rain. It will sloooowly degrade over time and eventually disappear.
Put it outside (recommended) somewhere and start it on fire (molotov cocktails come to mind, but there are other, odder, options. Killing a pet boomrat?)