Start with a base
Build a flat base first, like a stone brick pad with a polished andesite top. It makes every statue look finished, and you can measure the rest from it. A gold block plaque on the front is a nice touch.
Block out the big shapes
Place the body, head and legs as plain boxes before you add any detail. If the plain shapes already look like the animal from far away, the small parts will look good too. Fix the proportions now, not later.
Pick the right colour blocks
Concrete has flat, clean colours, which is why it works so well for statues. Use orange and white concrete for a cat, white for a unicorn, and quartz or smooth stone for a marble look. Wool also works and is easy to get in survival.
Give it a face
Eyes are what make a statue feel alive. Use black blocks for eyes, add a white block above each one for a sparkle, and put a pink block for the nose. Rosy cheeks make a statue look friendly.
Make it taller than you think
A statue should be bigger than a player so it feels special. Aim for at least twice your height. Giant statues are often easier, because you have more blocks to show the shape.
Light it up
Put lanterns on fence posts around the base so the statue is easy to see at night. Light also stops monsters from spawning on the base. Sea lanterns hidden in the base glow under the statue.