Help understanding <g> for Z/mZ

[Convergence]

Hello. So today in class, we talked a bit about $<g>$ as the set of all integer powers of $g$. Made enough sense. Then we did some examples in $\mathbb{Z} / 7\mathbb{Z}$, and I got a bit lost. I think this is more due to the fact that perhaps I don't quite grasp $\bmod{n}$ as an equivalence class. So I know that the elements of $\mathbb{Z} / 7\mathbb{Z}$ are $\{ [0], [1], [2], [3], [4], [5], [6] \}$, but we were looking for $<g>$ where $g = [5]$. The idea was we were showing that $\mathbb{Z} / 7\mathbb{Z}$ is cyclic. Since we were "brute-forcing" it so to speak, we wrote down elements in a certain order, which I have written down as $\{ [0], [5], [3], [1], [6], [4], [2] \}$ and then it cycled back to $[0]$.
So I suppose I'm having trouble doing computations in $\mathbb{Z} / 7\mathbb{Z}$, or $\mathbb{Z} / n\mathbb{Z}$ in general. I mean, I know that if $a \equiv b \bmod{n}$ then $n \mid a-b$, but in this case I'm not sure where to take it. Is it:
$5^0 = 1 \equiv$ something $\bmod{7}$, then $5^1 = 5 \equiv$ something $\bmod{7}$ etc?
Sorry if my question(s) is a bit hard to read; perhaps I didn't explain it as best I could. But any help would be appreciated.

 

[Convergence]

NEVER MIND!

Sorry, it's late, but I totally forgot the definition of $<g>$ when the group operation is addition. I computed it for that, and my results matched. I guess my problem is not with the computation, but rather when I need to take a break and go to sleep. I'm not sure I can close/delete this thread, so I'm posting this just in case.