Find all entire functions $f$ such that for all $z\in \mathbb{C}$, $|f(z)|\ge \frac{1}{|z|+1}$

Hint: As you suggest, you can consider $g = 1/f$, which is itself an entire function (why?). Use the estimate $|g(z)|\leq 1 + |z|$ and the Cauchy estimates to show that $g$ is a (nonvanishing) polynomial of degree $\leq 1$. What can you conclude?

Suppose $f$ is not constant. As an entire non-constant function it must have some sort of singularity at infinity. It cannot be a pole (because then it would have a zero somewhere, cf the winding-number proof of the FTA), so it must be an essential singularity. Then $zf(z)$ also has an essential singularity at infinity. But $|zf(z)|\ge \frac{|z|}{|z|+1}$, which goes towards $1$, which contradicts Big Picard for $zf(z)$.