Answer
$\nabla f = \left \langle \frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}} \right \rangle$. See graph.
Work Step by Step
We know that the gradient is just the partial of both parts. $\nabla f = \left \langle \frac{\partial} {\partial x} (\sqrt{x^2 + y^2}), \frac{\partial}{\partial y} (\sqrt{x^2+y^2}) \right \rangle = \left \langle \frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}} \right \rangle$
