As Radhil pointed out in a comment, might it be The Magician's Nephew?
Reading the synopsis on Wikipedia, apparently there is a lot more in the book than I remember reading long ago, but I distinctly remember the main characters jumping in pools to transport to different worlds. And, being a part of the Narnia series, there are definitely talking animals.
From Wikipedia's synopsis:
Digory finds himself transported to a sleepy woodland with an almost narcotic effect; he finds Polly nearby. The woodland is filled with pools. Digory and Polly surmise that the wood is not really a proper world at all but a "Wood between the Worlds", similar to the attic that links their rowhouses back in England, and that each pool leads to a separate universe. They decide to explore a different world before returning to England, and jump into one of the nearby pools. They then find themselves in a desolate abandoned city of the ancient world of Charn. Inside the ruined palace, they discover statues of Charn's former kings and queens, which degenerate from the fair and wise to the unhappy and cruel. They find a bell with a hammer, an inscription inviting the finder to strike the bell.
I do not remember anything about a fisherman nor a bone necklace/bracelet.