Thinking of a cube as $\mathbb Z_6\times\mathbb Z_4$ I was wondering if it makes sense to think of a cube as an additive group in this way. For example $(4,1)$ corresponds to face $4$ edge $1$.
I am a beginner in group theory and I hope this makes sense!
If I added together face $4$ edge $1$ to face $3$ edge $0$: $(4,1)+(3,0)=(1,1)$ would correspond to face $1$ edge $1$?
 A: A possible problem with treating the cube as $\mathbb{Z}_6 \times \mathbb{Z}_4$ is making sense of what that allows you to do.  In your question, at the end, you note that you could add "face $4$, edge $1$" to "face $3$, edge $0$" and get "face $1$, edge $1$".  But what does that actually mean?
If we label our cube so that each face has a number like a traditional die then opposite pairs of faces add to 7 (so 1 is opposite 6, 2 is opposite 5 and 3 is opposite 4).  So your representation allows you to "add" edges of faces that aren't connected to each other.  Is that useful?  (It might be, by the way, but I can't think of a use at the moment.)
So: either we need to add some more structure to your representation in order to "forbid" additions that don't make sense, or we need to change the representation in order to ensure that the operations we can perform all make sense.  Both are valid approaches: we could say for example that we can only "add" elements of the group together if they share a common edge or a common face.  You'll come across this in your course when you start talking about generators and relations which essentially say: here are specific relationships, and these are the permissible operations on them.
The dihedral group $\mathbb{D}_n$ was mentioned by a commentor above: this takes the symmetries of the square as the essential elements: rotations and reflections, and then specifies how they fit together.  If $a$ is rotation by $90^\circ$ about the $y$-axis and $b$ is reflection in the $y$-axis then you can state that $a^4 = 1$ since four rotations take you back to where you start, and $b^2 =1$ since two reflections take you back to where you started.
As @user1729 notes in the comments, the symmetry group of the cube is more complicated, and you'll find it easier to start with the dihedral groups since you mention being new to group theory.  They also note that your group $\mathbb{Z}_6 \times \mathbb{Z}_4$ is abelian (i.e. given any two elements $x,y$ in your group you have $xy = yx$).  This means that the order in which you apply your operations doesn't matter.  In the dihedral group, and most symmetry groups, it does matter, and the groups are non-abelian.  You can check this easily yourself: draw a square and label the corners $1,2,3$ and $4$.  Then reflect it about the $y$-axis and rotate it $90^\circ$ clockwise and note where the labels end up.  Then start from the first square and rotate then reflect and see that the labels end up in different places.
A: While your approach isn't invalid, I want you to ask back. What can be achieved via your definition of cube group? Why is it useful? Do not take this as an offense because I didn't mean it. They are questions needed for mathematics.
You defined how to match an element of a cube to an element of $\mathbb{Z}_6 \times \mathbb{Z}_4$. There's basically no problem. Also, you said $(4,1) + (3, 0) = (1, 1)$ on $\mathbb{Z}_6 \times \mathbb{Z}_4$. That's correct. Then you linked it to an operation on cube: adding face $4$ edge $1$ to face $3$ edge $0$, to get face $1$ edge $1$. But what's meaning on that? You can address meaningful explanation of that operation? Maybe a geometrical meaning?
If you can present such a meaning: that will be very nice. Maybe the group will be named after you. Or unfortunately somebody else already did it.
For example describing structure of cube, is a symmetry group of the cube. This is a group about cube, or more strictly, rotations and reflections of the cube(and octahedron). An element of the group means one exact rotation and reflection(that means, kinda preserving faces', edges' and vertices' locations) of the cube. They are total 24 ways to rotate, and 2 ways to reflect. Also, group operation(multiplying two elements of group) means composition of such rotations and reflections. So it addresses something, namely, symmetric structure of a cube.
So you defined a cube group structure of your own. That's great. The next thing is, to find it useful. Maybe you cannot find it, but it's OK. That's how we do mathematics. Just keep trying.
A: I don't see how faces and edges are related here. I makes sense, but it doesn't seem to have anything to do with the cube. For instance, you could take four types of pens in six different colors. Then define addition of pen i with color j to be like the group $\mathbb{Z}_6 \times \mathbb{Z}_4$. ... If it helps you think about the definition of the group, then that is fine, but often when one defines a group based on a geometric object like a cube, one is thinking about some sort of geometry related to it, but sometimes it helps to visualize things better.
It might be better to think of the group as a bike lock with two numbers to pick from, the first number to pick from is from 0 to 5 and the second number to pick from is from 0 to 3 and how you add two combinations of numbers is to rotate them the number of times in each slot. This naturally captures the idea of the group and the definition of the group is deeply tied into the geometry/movement of the object used to explain it. 
For example, this Bike lock could be used to $\mathbb{Z}_{10} \times \mathbb{Z}_{10} \times \mathbb{Z}_{10} \times \mathbb{Z}_{10}$.
