Donating a certain amount will unlock new items in the shop, so it's just quantity, while it really doesn't matter what you're donating (so feel free to just give them all the junk items you'll never use). Check out the wiki to find out more in detail:
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In general, most of your new equipment will be found in dungeons (the exception being the items you'll unlock by donating a lot of junk in the shop eventually), so there's no need to farm unless you starting to feel underpowered. As you're completing quests the items will gradually get better and better. Fwiw, the game has a mechanic that increases the chance of getting a rare item: the more chests you open, the more likely you'll find a rare piece (it resets to 0 again once you get a rare item tho), so if you ever feel like farming, missions with a lot of chests, like Cheetah Mind or Malgrid 8 would help you get those rares faster.
Lastly, keep in mind that buying a weapon or armor from the shop increases their rank (from 0 to 9), so if you want better stats (+5% per rank) on the items you're currently finding, you can always consider buying that particular weapon/armor in the shop to increase its rank throughout the game (assuming the item is available in the shop, not all of them are).