How do we conclude that following two groups are isomorphic? Let $H= \Bbb Z  \times \Bbb Z \times \{0\}$ and $T = \{0\} \times \Bbb Z \times \{0\}$
Then $H/T $ is isomorphic to $\Bbb Z$
$\Bbb Z$ denotes integers, and $\times$ Cartesian product.
How did we make that conclusion? I come across such examples very often, could you give me some tips in order to see such I believe "triviality"?

The trick is you can just divide, so $\Bbb Z / \{0\}$ becomes $\Bbb Z$ and so on, and $\Bbb Z \times \{0\} \times \{0\} \simeq \Bbb Z$. A formal proof would be along these lines:
$(a, b, 0) - (c, d, 0) = (0, e, 0) \iff a = c$, so the cosets would be of the form $\{a\} \times \Bbb Z \times \{0\}$ for each $a \in \Bbb Z$, so the quotient is isomorphic to $\Bbb Z$
