As a profession, you have to have a strong background in biology and geology. The usual route is to go to college (university) and get a degree in either biology or geology, with lots of electives in the not-major subject. Some places even offer a bio-geo degree (sort of a double degree), or used to anyway.
As a practical matter, you can start now but becoming a fossil hound and learning from experience. Most of the really good paleontologists I have met were folks who just loved it and always did it since they were young (your age or younger). Fossil nerds, in a way.
The problem, of course, is that geology is not the same everywhere, and some regions are way better for finding fossils than others, and if you don't live in an area where fossils are easy to find just by checking the local outcrops, you can look all you want but you won't find them.