Formula for the square root of a number?

There is not. The numbers that can be formed from rational numbers using a finite sequence of $+,\times,-$, and $\div$ are rational numbers, but many square roots of rational numbers ($\sqrt{2}$ for example) are irrational.

However, with an infinite number of steps it is possible. See Methods of computing square roots. A popular one is the Babylonian method: $$ x_{n+1} = \frac{1}{2}\left( x_n + \frac{x_0}{x_n} \right) $$ Let $x_0$ be the number you want to find the root of. Iterating this formula results in a sequence $x_0, x_1, \dots$ which approaches $\sqrt{x_0}$.

Back in the Olden Days, when I was in school, we learned an algorithm for square-root extraction similar to the long division algorithm. But that topic is long gone: there is no need to learn to do this by hand, since computers and calculators do it quite well.

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Here is the method.

Examples from Wikipedia:

No, because square root is not a rational function.