Betta fish are the only fish even remotely appropriate for a one-gallon tank. Because they are anabantids they can breathe air from the surface, and they can survive in less-than-ideal conditions. Small, one-gallon tanks often have poor filtration, and they pollute quickly, meaning most fish will perish.
But even the betta isn’t really a great choice. In fact, in my opinion, no fish should ever be kept in a one-gallon tank. I suggest at least a 5-gallon tank or bigger for bettas, and I would prefer you choose a 10-gallon or bigger for every other species. It gives fish enough room to swim, and it gives you better options for heat and filtration.
It isn’t possible to learn how to care for an aquarium properly with a tiny tank. Bigger tanks are easier to care for, and give you a greater chance at success as a new aquarium owner.
You may be tempted to consider a goldfish for a one-gallon tank, but they are not appropriate either. All varieties of goldfish grow much too large for such a small aquarium. They also pollute the water very quickly in a confined space. Goldfish should be kept in very large tanks or ponds.