Can anyone guide me in the general direction of the answer to the following:
A die is rolled $n$ times
$$A = \text{no $6$s}$$
$$B = \text{no $5$s}$$
$$P(A\cup B) = \;?$$
I am first finding $P(A)$ and $P(B)$ as (where $n$ = 10):
$$P(A)=\left(\frac 56\right)^{10}$$ $$P(B)=\left(\frac 56\right)^{10}$$
$$P(A\cup B)=P(A)+P(B)$$
Is this correct or there is some intersection I have to worry about?
Yes, you have to worry about the intersection. Use the formula $$P(A \cup B)=P(A)+P(B)-P(A \cap B).$$ In your case $$P(A\cap B)=(4/6)^{10}.$$