As
$$
f(\mathbf{x}):=||\mathbf{x}||_{2}^{p}=\left(\sum_{i=1}^{n}x_{i}^{2}\right)^{p/2}
$$
and
$$
\nabla f(\mathbf{x})
:=
\left(\frac{\partial f}{\partial x_{1}},\frac{\partial f}{\partial x_{2}}, \ldots, \frac{\partial f }{\partial x_{n} } \right)
$$
and noting that
$$
\frac{\partial f}{\partial x_{j}}
=
\frac{p}{2}\left(\sum_{i=1}^{n} x_{i}^{2} \right)^{p/2-1}\cdot 2x_{j}
=px_{j}||\mathbf{x}||_{2}^{p-2}
$$
then
$$
\nabla f(\mathbf{x})=p||\mathbf{x}||_{2}^{p-2}\left(x_{1},x_{2},\ldots, x_{n}\right)
$$
As for the Hessian,
$$
\nabla^{2}f
:=
\begin{pmatrix}
\frac{\partial^{2} f}{\partial x_{1}^{2}} &
\frac{\partial^{2} f}{\partial x_{1} \partial_{x_{2}}} &
\cdots
&\frac{\partial^{2} f}{\partial x_{1}\partial_{x_{n}}} \\
\frac{\partial^{2} f}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f}{\partial x_{2}^{2}} & \cdots & \frac{\partial^{2} f}{\partial x_{2}\partial_{x_{n}}} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial^{2} f}{\partial x_{n} \partial_{x_{1}}}
&
\frac{\partial^{2} f}{\partial x_{n} \partial_{x_{2}}}
&
\cdots
&
\frac{\partial^{2} f}{\partial x_{n}^{2}}
\end{pmatrix}
$$
so we consider two cases: The diagonal elements and off-diagonal elements. These entries are computed easily from standard rules of calculus; I'm too worn out to compute them explicitly.
A good book on vector calculus is Div, Grad, Curl, And All That: An Informal Text on Vector Calculus by H.M. Schey. Newton's Method in the multivariate case is a pretty straightforward generalization of the single-variable case, noting that
$$
f(\mathbf{x}+\mathbf{h})\approx f(\mathbf{x})+\left<\nabla f(\mathbf{x}),\mathbf{h} \right> + \frac{1}{2}\left<\mathbf{h},\nabla^{2}f(\mathbf{x})\mathbf{h}\right>
$$
where $\left<\cdot, \cdot\right>$ denotes the ordinary dot product on $\mathbb{R}^{n}$.