What is the study of fungi called?

Mycology, and fungi are not plants - they are closer to animals. They cannot photosynthesise, they digest their food externally by secreting enzymes and absorbing what dissolves, and their cell walls are made of chitin, the same material as an insect's shell. The split from the animal lineage came after both had already separated from plants.

Most of a fungus is invisible. The mushroom is only the fruiting body; the organism itself is a network of threads through soil or wood, and it can be enormous - a honey fungus in Oregon covers some nine square kilometres and is estimated at two to eight thousand years old, which makes it a serious candidate for the largest and oldest living thing on the planet.

They also hold up the terrestrial world in ways that get little credit. Fungi are the principal decomposers of wood; around ninety per cent of land plants depend on fungal partners at their roots to obtain water and phosphorus; and the arrangement appears in the fossil record at the same moment as the first land plants, which suggests plants only managed to leave the water with fungi already attached. Penicillin, statins and the immunosuppressant that made organ transplants possible all came out of fungi too.