Find the order of $(6,15,4)\in \Bbb{Z}_{30}\times \Bbb{Z}_{45}\times \Bbb{Z}_{24}$. 
Find the order of $(6,15,4)\in \Bbb{Z}_{30}\times \Bbb{Z}_{45}\times \Bbb{Z}_{24}$.

Since $|6|=5$, $|15|=3$, and $|4|=6$, then $|(6,15,4)|=\operatorname{lcm}(5,3,6)=30$. Is this correct? I found the order of $6$ by adding $6$ until it was congruent to $0$, making $\left \langle 6 \right \rangle$ have $5$ elements. However, I am unsure whether that is the right way to find the order of each element.

Everything you did is correct.
Here is a formula for the order of an element in $\mathbb{Z}_n$ if you wish to be sure:
$$|k| = \frac{n}{\gcd(n,k)}$$
