If you want to make green dye in Minecraft, you will need to find a cactus block. That said, cactus blocks are spikey plant blocks that can be found primarily in desert biomes and badlands. So if you want to find one, you will need to start in either of these biomes.
However, if you had to choose where to search, we would highly recommend the desert over the badlands. The reason being is that cactus blocks are far more common in the desert biome.
Alternatively, you can also search villager houses and even purchase them from wandering traders with emeralds. If you don't get these emeralds, you can always try mining down near diamond level to get them.
That being said, once you found a cactus block, you can turn it into green dye by placing the cactus inside a furnace and letting it burn. After a few seconds, the cactus will turn into green dye, but you need to make sure you have both a furnace and some fuel before you can melt down cactus blocks.
You can get a furnace with eight stones placed in every slot of a crafting table but the middle, and you can use wood, coal, and charcoal as good starting fuel.
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