Vertebrate Paleontology - Romer - was created in 1933.

Journal of Vertebrate Paleontology was created in 1980.

Vertebrate paleontology, Invertebrate paleontology, Paleobotany, Micro-paleontology, Palynology, Economic paleontology, Applied paleontology, Paleozoology, Paleoecology, Paleolimnology

of course vertebrate paleontology.
vertebrates are all the animals with a backbone.
invertebrates are all the animals without it (e.g. shells, coralls, sponges, cuttlefish...).
botany has nothing to do with any animals, but with plants.

You should get a degree in Earth Sciences or Geology, many offer courses in invertebrate and vertebrate paleontology.