There is not a lot of plot to this -- it is mainly an excuse to have a few funny tall tales about a gambler in the West in the late 1800s.
The narrator, Mark Twain, goes looking for a man named Leonidas Smiley. When he asks about Smiley, a man in a bar starts telling him about some other guy called Jim Smiley who was an eccentric gambler. He tells Twain about all the crazy things this guy would do to gamble.
The longest story is about how Smiley had a champion jumping frog that he would bet on. Someone got the frog to swallow a bunch of lead shot to weigh it down and so Smiley lost his bet.
Eventually, Twain gets tired of hearing these stories and leaves without finding out about Leonidas Smiley.
The story is a frame story, that is, a story within a story. As such, it has two plots. The outside story, the frame, concerns the unnamed narrator's quest to find a man named Leonidas Smiley at the behest of a friend. It turns out that the narrator is the butt of a practical joke, for in advising the narrator to look up Leonidas Smiley, his friend knows he will be sucked into a lengthy conversation with the garrulous Simon Wheeler. The goal of the main character, the narrator, is to get away without wasting any more of his time on this wild goose chase. The plot is resolved when the narrator manages to extract himself from Wheeler's long-winded one-sided conversation.
The second story, the inside story, is told by Simon Wheeler. This story's plot revolves around the goal of the main character, Jim Smiley, to win at gambling. Jim Smiley is hopelessly addicted to gambling. The rising action of the story tells about Smiley's increasingly complicated betting schemes. He bets on raindrops; on the death of the preacher's wife even though she has recovered from an illness; on his "fifteen-minute nag"; on his bull-pup, Andrew Jackson; and finally on his frog, Dan'l Webster. The climax comes when Jim Smiley is snookered by a stranger who fills Dan'l Webster with buckshot on the sly, causing Smiley to lose the bet. The resolution of the plot is that Jim Smiley gets bested by the stranger and never catches him.
To understand the plot of "The Celebrated Jumping Frog of Calaveras County," one needs to understand that it is two stories in one and to consider each plot separately.