How to you build a moving ship in Minecraft?

Pistons can be triggered by a redstone signal, and will push blocks in front of them. Slime blocks, when pushed by a piston, will drag other blocks next to them alongside them—and if you put an observer (or, with more finesse, a redstone block) just right, you can make it so that when the piston triggers, it pushes a observer/redstone block into another piston, which in turn triggers. (For example, put a redstone block, or an observer with the red dot facing to the right in this diagram, like this: ...SSSQ P where the S is slime, Q is the redstone block or observer, and P is a piston.) Each piston should only try to push 12 blocks at a time—more and it will not extend.

If you wind this chain of pistons around on itself (you need at least 3 to avoid block tick screwiness, I don't know why, I just know that when I try it it never works with less than 3), you can make it continue to push itself forever—this is one of the simplest flying machine engines. You can then use more of the piston-observer/redstone-block structures (a flying machine extender), and more slime to attach aesthetically appealing structures to the core engine, as well as cubbies/hooks to hold onto yourself and entities. So, you can make your flying machine look like a boat on the waters.

Making the flying machine go both ways can't be done with the engine above, so you'll need fancier ones. They all still operate on the same technique, however—pistons that ultimately, through a lot of indirection, push themselves.

(I will note that in a comment, you say you want to do this not only for appearances but also for extra storage—but if all you wanted was extra storage, you could just lead a bunch of donkeys/llamas and bring them across the water with you. The flying machine is solely to look cool, and is otherwise utterly impractical.)