I am just looking for a bit of help understanding this concept: Let $$\mathbb Z/6\mathbb Z=\{[0],[1],[2],[3],[4],[5]\}$$be the set of equivalence classes.
But what elements does $\mathbb Z/6\mathbb Z$ under addition modulo $6$ have? Is it again the same as $\mathbb Z/6\mathbb Z$ or is it $\mathbb (Z/6\mathbb Z,+6)=\{0,1,2,3,4,5\}$
I am having trouble differentiating between the set and the set under the group operation.
You can write $[n]$ to underlie that you're working with equivalence classes (and this is the "correct" way to write the elements of a quotient) but to simplify the notation, sometimes you can also just write $0,1,\cdots,5$ for the classes $[0],\cdots,[5]$ if there's no risk of confusion with actual integers and remember that, for example, $3+5=2$ in this case. The result is a different, although equivalent ring. As Bill Dubuque points out, we choose $0,1,\cdots,5$ as representatives because they are the "minimal" representatives, meaning that every other $n\in\mathbb Z$ can uniquely be written as $6q+r$, with $0\leq r\leq 5$.