How do multiply the nabla operator by $f$? I have the function $f=\{x,-y,0\}$, and each of the derivatives together result in $\{1,-1,0\}$. To calculate the divergence $\nabla\cdot$ of $f$ I'd have to do the dot (scalar) product of partial derivatives with $f$, while for the curl $\nabla\times$ I have to do the cross product. I
I know that the curl is $\{0,0,2\}$ but I do not really understand how this result comes up: the cross product of the partial derivatives with $f$ results in something like $\{0,0,x+y\}$. 
Please bear in mind I'm very new to this topic and no one has taught me this, so it would help if someone could point out what has to be multiplied by what exactly. Thanks.
 A: So what I understand is that you have trouble calculating the curl of a given vector field
$$
F_x = x\qquad F_y=-y \qquad F_z = 0
$$
In Wikipedia, for example, you can find the quite straightforward formula for the curl: 
$$
\nabla\times F = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\ \right) \hat{i}
+\left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\ \right) \hat{j}
+ \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\ \right) \hat{k}
$$
As you see, there are no terms $\frac{\partial F_x}{\partial x}$ or $\frac{\partial F_y}{\partial y}$, so therefore the curl is zero.
A: Let's start from the definitions. We have a vector field (vector-valued function) $$\bbox { \begin{aligned}
\mathbf{F}(x, y, z) &= \bigr ( X(x, y, z) , Y(x, y, z) , Z(x, y, z) \bigr ) \\
\; &= X(x,y,z) \hat{\mathbf{e}}_x + Y(x, y, z) \hat{\mathbf{e}}_y + Z(x, y, z) \hat{\mathbf{e}}_z \\
\end{aligned} }$$
where $X(x, y, z)$, $Y(x, y, z)$ and $Z(x, y, z)$ are scalar functions, and $\hat{\mathbf{e}}_x$, $\hat{\mathbf{e}}_y$, and $\hat{\mathbf{e}}_z$ are the standard unit vectors in the directions of the $x$, $y$, and $z$ coordinates, respectively.
Divergence is defined as
$$\bbox {
\nabla \cdot \mathbf{F}(x, y, z) = \frac{\partial X(x, y, z)}{\partial x} + \frac{\partial Y(x, y, z)}{\partial y} + \frac{\partial Z(x, y, z)}{\partial z}
}$$
and curl is defined as
$$\bbox { \begin{aligned}
\nabla \times \mathbf{F}(x, y, z) &= \left [ \begin{matrix}
\hat{\mathbf{e}}_x & \hat{\mathbf{e}}_y & \hat{\mathbf{e}}_z \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
X(x,y,z) & Y(x,y,z) & Z(x,y,z) \\
\end{matrix} \right ] \\
\; &= \hat{\mathbf{e}}_x \left (
\frac{\partial Z(x,y,z)}{\partial y} - \frac{\partial Y(x,y,z)}{\partial z}
\right ) \\
\; &\, + \hat{\mathbf{e}}_y \left (
\frac{\partial X(x,y,z)}{\partial z} - \frac{\partial Z(x,y,z)}{\partial x}
\right ) \\
\; &\, + \hat{\mathbf{e}}_z \left (
\frac{\partial Y(x,y,z)}{\partial x} - \frac{\partial X(x,y,z)}{\partial y}
\right ) \\
\end{aligned} }$$
These are the same things if you wrote $\nabla = \left ( \frac{\partial}{\partial x} , \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right )$ and did the dot and cross products, respectively.

In OP's case, $X(x,y,z) = x$, $Y(x,y,z) = -y$, and $Z(x,y,z) = 0$.
For divergence, that gives us
$$\bbox{ \begin{aligned}
\nabla \cdot \mathbf{F}(x, y, z) &= \frac{d x}{d x} + \frac{d(-y)}{d y} + 0 \\
\; &= 1 - 1 \\
\; &= 0 \\
\end{aligned} }$$
For curl, we have
$$\bbox{ \begin{aligned}
\nabla \times \mathbf{F}(x, y, z) &= \hat{\mathbf{e}}_x \bigr ( 0 - 0 \bigr ) \\
\; & + \, \hat{\mathbf{e}}_y \bigr ( 0 - 0 \bigr ) \\
\; & + \, \hat{\mathbf{e}}_z \bigr ( 0 - 0 \bigr ) \\
\; & = \bigr ( 0, 0, 0 \bigr ) \\
\end{aligned} }$$

Note that Maple agrees,
>  with(VectorCalculus):
>  f := VectorField(<x,-y,0>, 'cartesian'[x, y, z]):
>  Divergence(f);
        0
>  Curl(f);
           _       _       _
        (0)e  + (0)e  + (0)e
            x       y       z

and so does SageMath:
sage:  var('x y z')
sage:  f = vector([x, -y, 0])
sage:  f.div([x, y, z])
0
sage:  f.curl([x, y, z])
(0, 0, 0)

They are rather nice tools to use to verify your calculations.
Note that I myself am not a mathematician, but use math as a tool constantly.  Even though I use Maple and SageMath almost exclusively to do the hard work, it only works if I know the rules and operations and methods.  Maple and SageMath and other tools are definitely not a replacement for learning the math; but when you do have a grasp on the math, they do save a lot of time, and help you avoid most errors. (They do not help you avoid logical errors, like applying the wrong tool or algorithm or solution method to the wrong problem; which is why there is no alternative to learning math.)
