Specific example of a quotient group I am trying to self study groups, and may be stuck on this definition:

The quotient group has group elements that are the distinct cosets, and a group operation $(g_1 H)(g_2 H) = g_1 g_2 H$
  where $H$ is a subgroup and $g_1,g_2$ are elements of the full group $G$.

Let's take this example:
$G$ is the group of integers, with addition.
$H$ is the group of integers divisible by 3 also with addition,  -3,0,3,6,9,....
What is the quotient group for this case?
I take the definition to mean that the quotent group elements are not single integers, but infinite sets of integers, the cosets. 
The cosets are ${i + H, i \le 3} = 0+H,1+H,2+H = (...,0,3,6...), (,1,4,7...), (,2,5,8…)$
so the quotient group has 3 elements.  Is this correct?
Then what is $G/H$?  
From the definition above it seems that an element of the quotient group should be 

$G/H = g_1 + g_2 + H =  g_1 + g_2 + (-3,0,3,6,9,…)$

i.e. it is the subgroup H with two arbitrary integers added to each element.
 A: If $G=\langle \mathbb Z, +\rangle$ so, it is abelian infinite group and every subgroups of $G$ is normal in it so we can speak about $G/H$ freely. Here, you took $H=\langle 3\rangle=3\mathbb Z$, so we have $$G/H=\{0+H,1+H,2+H\}$$ because if we divide any integer numbers by $3$, the reminders are $0,$ $1$ or $2$ and the rest would be put inside $H$. This means that $H$ eats a considerable part of any integer numbers but three numbers $0,$ $1$ and $2$. You may have seen that every finite group of order $3$ can be written as $$\mathbb Z_3=\{0,1,-1\}$$ Now you can set $0\in\mathbb Z_3$ with $0+H\in G/H$ in your mind and think of $G/H$ as $\mathbb Z_3$. Note that this imagination is not a Mathematical way but here you are allowed to do that. I hope you read about isomorphisms later ann then do this final part with a simple map. 
A: $G/H$ is a group with $3$ elements. 
A theorem in group theory states that a group $G$ with $|G| = p$, $p$ prime, is cyclic. So $G/H$ is the cyclic group with $3$ elements.
