The group: $\dfrac{(\mathbb Z/2 \times \mathbb Z/4)}{\langle (\bar{1}, \bar{2}) \rangle} $.
I understand what the elements of the product group in the numerator looks like, but I don't understand what the elements in the group look like of the above.
I have to find an isomorphic group to this that is well known. I think I can do this by calculating cosets, but again I am not sure how.
Hint: Compute the order of the subgroup generated by $(\overline{1},\overline{2})$ (i.e. the order of this generator). How many elements can the quotient have? This will already tell you a lot in this case.
How do the elements of the quotient look like? Well they are tuples of residue classes mod $2$ and $4$ that are identified with the zero tuple whenever it is a multiple of $(\overline{1},\overline{2})$. You basically kill all multiples of $(\overline{1},\overline{2})$ and keep the rest (just respect the new relation that the quotient gave you).
So an example of such an element would be $(\overline{1},\overline{3}) + \langle (\overline{1},\overline{2}) \rangle$. This is just done in the same way as $\overline{2} = 0 + 2\mathbb{Z}$.
You can of course also use another overline to indicate that you are in this new quotient instead of writing the summand on the right, but that gets confusing in my opinion.