For $p=2$: up to multiplicative universal constants, the average $M$ of  $\|\cdot\|_q$ over $S^{n-1}$ is equal to 

 1. $M \simeq \sqrt{q} \cdot n^{1/q-1/2}$ when $1 \leq q \leq \log n$,
 2. $M \simeq \sqrt{\log n}/\sqrt{n}$ for $q \geq \log n$.

This can be checked most easily after switching to a Gaussian integral as Yemon mentions. For the lower bound in 1, it may be useful to consider using concentration of measure. If it is a matter of reference, it can probably be extracted from Chapter 5.4 in Milman-Schechtman, "Asymptotic theory of finite-dimensional normed spaces". Indeed, the value of this average is closely related to the dimension of almost Euclidean sections of the space $\ell_q^n$.