As a geoscientist, I can address this question.

If the dinosaurs that you're interested in are the big extinct animals like T. rex and Triceratops (and not birds), then you're solidly in the realm of Earth history, rather than the human history discussed in this subreddit. The history of the Earth is a fundamental part of Earth science (a.k.a. geoscience), and it's required learning for anyone entering the field, since an overview of Earth history gives you the time and environmental context for just about anything else you might want to learn.

Much of early paleontology work focused on excavation, description, and classification. But within the context of an Earth history that these days is much better understood, paleontology as a sub-discipline now also brings in elements of biology, chemistry, and physics. We can ask questions about how organisms lived and reproduced, how they moved (and how fast), what habitats they preferred, and sometimes even what social behaviors they might have engaged in. Your interest in a specific kind of research question would govern what other sciences you'd have to tap in greater depth in order to look for answers. For example, figuring out how dinosaurs walked/ran involves biomechanics, which requires physics.

To bring this question back into the time frame of human history: Dinosaur paleontology as a profession is largely in the realm of academia and museums. It's also a rather different pursuit nowadays compared to the eras of most of the major finds: the "Bone Wars" of O.C. Marsh and Edward Drinker Cope from about 1870-1897, where the two men were in a no-holds-barred rivalry to discover as many dinosaurs as they could in the western US; and later in the 1920s, the great fossil finds in Mongolia by Roy Chapman Andrews. Firstly, large numbers of fossils exist in museum collections that were never fully examined after collection, so significant scholarship is still possible just through museum-based work -- which is far less expensive than mounting field expeditions. Secondly, governments have generally become much more protective of their natural heritage, so official excavations are much more carefully controlled, and the fossils themselves usually need to remain in the country where they're found. (The latter measure has resulted unfortunately in a thriving black market for illegally exported fossils.)

An exception to this export control exists in the US, where finds on private land remain the property of the land owner. This has lead to auctions of various fossils, including dinosaur eggs and bones. Two T. rex skeletons from South Dakota currently hold the record for the highest prices brought at auction: Stan for $31.8 million in 2020, and Sue for $7.6 million in 1997.

I wouldn't consider dinosaur paleontology to be a field with a lot of career potential, but you can always consider pursuing it as a hobbyist in the U.S. If your particular interest is in excavation, your best bet is to first try to find a volunteer opportunity to join experienced researchers, to learn proper field techniques so that you don't damage any finds you may make.

In any case, I would start by digging into (no pun intended) some general dinosaur books - there are plenty out there! I would also seriously recommend learning some Earth history so that you can understand the context and importance of your finds.

References:

Jaffe, Mark (2000). The Gilded Dinosaur: The Fossil War Between E. D. Cope and O. C. Marsh and the Rise of American Science. New York: Crown Publishing Group. ISBN 978-0-517-70760-9. Available for loan at https://archive.org/details/gildeddinosaur00mark

Dixon, Dougal (2011). The Complete Book of Dinosaurs. Wigston: Southwater. ISBN 978-1-780-19037-2.