The items that work for Currency are listed on that page, so won't be covered here.
There are Quest Items that are gained either by purchase from Shops or by completing Quests (oddly enough).
Custom Items covers both your House and your Character.
While you can purchase a handful of these from Events, or find them via Treasure Maps, these are primarily gained by earning the rarely found Custom Items from Map Zones.
There are six categories here:
Only the Weapon gives benefits to your character, while the Pets will simply help you whack bushes a little faster. Everything else is purely cosmetic.