What are the units of $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$ I'm struggling with this a lot.
I think it all boils down to my basic understanding of what the ring   $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$ is. As much as I know is that it is a ring of 10 elements, but I have no idea how I would go about finding the units in this.

The way I view it is:

$\mathbb Z/2\mathbb Z  = \{0,1\},\qquad
  \mathbb Z/5\mathbb Z= \{0,1,2,3,4\}$.

So would   $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z= \{0\times0, 0\times1, 0\times2, 0\times3, 0\times4, 1\times0, 1\times1, 1\times2, 1\times3, 1\times4\} = \{0,0,0,0,0,0,1,2,3,4\}$?

This seems totally incorrect to me, so it would be great if someone could clear this up for me.

I understand a unit is an element $u$ such that $uv=vu$, but in this I don't really even know what the elements are...
Would the unit be $0$ and $1$?

Any help would be great.
Thanks
I'll clarify again the product of rings:
For $R, S$ rings, the ring $R \times S$ is defined to be
$$R \times S := \{(r,s) \mid r \in R, s \in S\}$$
with addition and multiplication defined pointwise, additive identity $(0_R, 0_S)$ and multiplicative identity $(1_R, 1_S)$ and additive inverses also defined pointwise.

So, the product $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$ looks like
$$\{(0,0),(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(1,4)\}$$
with $10$ elements as you expected.

Now, for $u=(u_1,u_2)$ to be a unit, we need to find a $v = (v_1, v_2)$ such that $uv = 1$. Looking at the definition of the product ring, this means $u_1 v_1 = 1$ and $u_2 v_2 = 1$, i.e. $u_1$ and $u_2$ are units in their respective rings. In $\mathbb{Z}/2\mathbb{Z}$, the only unit is $1$ and in $\mathbb{Z}/5\mathbb{Z}$, the non-zero elements $1,2,3,4$ are all units. So, $u = (1,1),(1,2),(1,3),(1,4)$ are the units of $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$.
Just make the multiplication table.

The ring $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$ is made up of all ordered pairs $(p,q)$, where $p \in \mathbb Z/2\mathbb Z$ and $q \in \mathbb Z/5\mathbb Z$. 

Multiplication is defined in the natural way: $(p,q) \times (p',q') = (p\times p',q\times q')$.

The units are all those elements with multiplicative inverses.

You want pairs, for which there is another pair with $(p,q) \times (p',q') = (1,1)$, remembering to reduce the first component modulo 2, and the second component modulo 5, e.g.

$$(1,4) \times (1,3) = (1\times 1, 4 \times 3) = (1,12) = (1,2)$$