Computing factor groups when groups are of finite order is no problem, but when the group is of infinite order I struggle. I have looked at many solutions and really(!) tried to understand them with no luck. In my attempt to understand this, I saw that there are different ways to go about this. Some look at the cosets, and others use the first isomorphism theorem. No matter how hard I try, I am not able to understand any of them. 
For example $( \mathbb Z \times \mathbb Z \times \mathbb Z\_8 )$ $/ <(0, 4, 0)> $ or $(\mathbb Z \times \mathbb Z \times\mathbb Z)$$/<(3, 3, 3)>$ or $(\mathbb Z \times \mathbb Z)$ $/ <(1 , 2)>.$ These are just examples from my book (and I know what the answer is going to be). 

My questions are; how does one go about to solve this (in the easiest, beginner friendly way)? I need a dummies step by step explanation, because I dont see the point of the steps taken.