Fish move left to right because they’ve lived in water from the beginning and happened to develop stronger muscles that do the side to side motion. They could have done up and down but nature doesn’t care, if something works it stays.
Dolphins and whales move up and down because they evolved from land mammals that walked on four legs. The biomechanics of walking on four legs make it so that the spine has to have a little bit of up and down sway, plus they developed strong muscles to keep the spine up and hold the weight of the belly. Those muscles are responsible for an up and down motion. So when the ancestors of dolphins went back into the water, they already had those nicely developed muscles, and it was easy to modify the tail (not the legs like other comments claim) to propel them forwards by making the already existing, up and down moving muscles bigger and stronger.
The back legs of dolphins became useless and eventually shrunk down until they disappeared because it cost the body too much energy to make and maintain those limbs. Some individuals still have tiny bones that are remains of the hip bones, found near the spine. The tail got longer but there are no bones in the flat part of the tail: that’s mainly fibrous tissue, kind of like the cartilage in your ears.
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