understanding $\mathbb{R}$/$\mathbb{Z}$

One proves that $\mathbb R/\mathbb Z$ is isomorphic to the group of unit-modulus complex numbers (let's call it $G$), which is a circle, isn't it?

Let's prove the isomorphism. Take $\varphi : \mathbb R \rightarrow G$ defined by $\varphi(\theta) = e^{2\pi i\theta}$. We have $\varphi(\theta + \theta') = e^{2\pi i(\theta+\theta')} = e^{2\pi i\theta}e^{2\pi i\theta'} = \varphi(\theta)\varphi(\theta')$ so this is indeed a homomorphism. $\varphi$ is surjective, and $\varphi(\theta) = 1 \Leftrightarrow 2\pi\theta = 2k\pi (k\in\mathbb Z)$, so $ker(\phi) = \mathbb Z$.

By the first isomorphism theorem, $\mathbb R/\mathbb Z \simeq G$.

As for your second question, try to picture this: take a 1x1 square sheet, and join the opposite edges so as to get a torus. $G'/N'$ is exactly the same construction: you identify the 'points' $(+\infty,0)$ with $(-\infty,0)$ and $(0,+\infty)$ with $(0,-\infty)$.

I hope this clarifies a bit!

You can also use the following nice facts. I hope you are inspired by them.

$$\mathbb R/\mathbb Z\cong T\cong\prod_p\mathbb Z(p^{\infty})\cong\mathbb R\oplus(\mathbb Q/\mathbb Z)\cong\mathbb C^{\times}$$