This is a real simple question.
If I have a group $Z_4\times Z_6$, is its order $24$ or $lcm(4,6)=12$. My teacher told me its $24$, which makes sense because there are $4$ cosets for $H=\langle(2,2)\rangle.$
I only ask this because I have also seen that the group $Z_{10}\times Z_{15}$ has order $30$ and same goes for $Z_{30}\times Z_5$.
My question is which is it?
I think you may be confused between the concepts of the order of a group and the order of an element in the group.
The order of a group is the number of elements in the group. If $G$ and $H$ are finite, then the number of elements in $G\times H$ is the product of the number of elements in $G$ and the number of elements in $H$: the order of $G\times H$ is the order of $G$ times the order of $H$.
You could also ask what the maximum order of any element in such a group is: the order of an element $g$ in a group is the smallest positive integer $k$ such that $g^k = e$. In the case of $\mathbb{Z}/m\times\mathbb{Z}/n$, the largest order of any element is in fact $\mathrm{lcm}(m,n)$, so that for example in $\mathbb{Z}/4\times \mathbb{Z}/6$, the maximal order of an element is $\mathrm{lcm}(4,6) = 12$.