Sell your rare minerals. They are useless outside of a few equipment upgrades, jewelry, and re-crafting the champion's gear.
In terms of monster parts, they're fine for crafting elixirs, if you're into that, but they're most useful for obtaining Mon, the only alternate currency in the game. If you have more than 100 of any monster part, sell down to that number. You'll be fine.
As for the Great Fairy materials, check to see how many springs you've unlocked, and then the tier of the gear you're trying to upgrade. If the tier is higher than the amount of springs you have, you can't craft it, but also there are some rare materials that drop infrequently from enemies. Remember, the likelihood of higher tier material drops goes up as you complete more Shrines.