Minecraft wiki entry for Amethyst_farming, associated text: Amethyst clusters will only drop 2 shards if harvested by a piston, however if you harvest it with a Fortune 3 pickaxe it can give you either 8, 12 or 16 shards per cluster mined. That means that unlike most farmable mobs, items and blocks, farming amethyst manually can yield more amethyst than if you farm it automatically.
All the stages of amethyst buds and clusters will only grow and mature to the next stage if planted on a budding amethyst block. Budding amethyst blocks can produce new amethyst shards, but can only do so if a side is unobstructed and does not have any pending amethyst buds growing, and, like most blocks, it has 6 sides.
Budding amethyst blocks itself are non-obtainable blocks, like monster spawners, which means that they won't drop if you mine them, no matter what tool and which enchantments you use. It can only generate in amethyst geodes, so if you want to farm them, you need to find an amethyst geode, and turn that area into the farm.
A block of budding amethyst has a 20% chance every game tick to produce a small amethyst bud if there is a free slot on a side, however amethyst buds themselves take a few Minecraft days to mature. Amethyst geodes are normally in fairly remote locations, especially if you are playing on a world you created before 1.17, so it is recommended you set up a chunk loader there and come back when you want to harvest them.