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 $1$?

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 $\mathbb{Z}_{10} \times \mathbb{Z}_{10} \times \mathbb{Z}_{10} \times \mathbb{Z}_{10}$.
