Normalization of a vector is simply making a vector with the same direction, but of size 1. Thus if you you have a vector $\mathbf v$ and want to find the normalized vector $\hat{\mathbf v}$, you can calculate it by $\hat{\mathbf v} = \frac{\mathbf v}{|\mathbf v|}$ where $|\mathbf v|$ is the norm of $\mathbf v$.
I suspect your question is more about orthogonalization or maybe the concept of orthogonality than normalization. Though a bit of confusion maybe has come from that a vector is called normal to a subspace (the vector is called a normal vector to the subspace) if it is orthogonal to all vectors in the subspace.
The Gram-Schmidt procedure for only two vectors $\mathbf v_1$ and $\mathbf v_2$ can be applied as follows. First, set $\mathbf u_1 = \mathbf v_1$ and
$\mathbf u_2 = \mathbf v_2 - \frac{\langle \mathbf v_2, \mathbf u_1 \rangle}{\langle \mathbf u_1, \mathbf u_1 \rangle} \mathbf u_1$
where $\langle \mathbf v_2, \mathbf u_1 \rangle$ is the dot product of $\mathbf v_2$ and $\mathbf u_1$.
Now, this will give you two vectors $\mathbf u_1$ and $\mathbf u_2$ which are orthogonal, but not normalized. Use the normalization formula above to get an orthonormal system.
If you want to, you can apply the normalization to $\mathbf u_1$ before calculating $\mathbf u_2$. This way, you can skip the denominator in the formula, since if $\hat{\mathbf u}$ is normalized, $\langle \hat{\mathbf u}, \hat{\mathbf u} \rangle = 1$. You will however need to normalize $\mathbf u_2$ when you have calculated it.
For the second part, you want to find a three-dimensional vector $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ such that the dot product with a vector $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ satisfying $3x - 2y + z = 0$ is zero. Calculate the dot product. Can you identify $a, b, c$? When you have found $a,b,c$, normalize the vector.