Showing that the union of two Cantor sets is perfect is just a direct application of the definition of "perfect." The other claim is a bit harder.
Towards containing no intervals of positive length, here is a hint: Try to prove the following strengthening of "Cantor sets don't contain intervals of positive length":
Suppose $X$ is a Cantor set, and $I$ is an interval of positive length. Then there is a sub-interval of positive length, $J\subseteq I$, which is disjoint from $X$ - that is, $J\cap X=\emptyset$.
(Note that this isn't trivial: $\mathbb{Q}$ doesn't contain any interval of positive length, but there are also no intervals of positive length disjoint from $\mathbb{Q}$. You'll need to use some special property of Cantor sets . . .)
If you can prove this, then: suppose $I$ is an interval of positive length, and $C_0, C_1$ are Cantor sets. Pass to a $J_0\subseteq I$ disjoint from $C_0$, and then a $J_1\subseteq J_0$ disjoint from $C_1$, each of positive length. What can you say about $J_1\cap C_0\cup C_1$?