I won't say anything here about the number of components in your restricted representation, but here is some information about multiplicities.

Kwangho Choiy and Dipendra Prasad have (independently) formulated a conjecture that expresses multiplicities in terms of the enhanced Langlands parameter attached to your representation $\pi$. Choiy proves the conjecture for tempered representations under the assumption that the Langlands correspondence exists and has lots of expected properties. Prasad and I reduce the conjecture to the tempered case, give a heuristic for why it should be true, and compute some examples of multiplicity. In some sense, both papers are thus trying to answer your question. However, I don't know if the answer provided is useful for you, since one doesn't usually know the Langlands correspondence in an explicit way. So let me outline a few examples, and you can find details for most of them in the papers above or here.

Let $G'$ be a subgroup of $G$ that contains the derived group, and $F$ the local field. I'll assume that $F$ is nonarchimedean, but won't assume $F=\mathbf{Q}_p$. Let $Z$ denote the center of $G$. If the quotient $G(F)/Z(F)G'(F)$ is cyclic, then the multiplicity is $1$ for elementary reasons. However, this doesn't happen all that often, so one must use other techniques. Here are some pairs $(G,G')$ for which restriction from $G(F)$ to $G'(F)$ is always multiplicity free:

- $(GL_n,SL_n)$,
- $(GO_n,O_n)$ (using any nondegenerate symmetric form, and assuming $p\neq 2$),
- $(GSO_n,SO_n)$ (ditto),
- $(GSp_{2n},Sp_{2n})$,
- $(GU_n, U_n)$ (any nondegenerate hermitian form),
- $(U_n, SU_n)$ if $p$ is coprime to $n$, or $F=\mathbf{Q}_p$ with $p\neq 2$,
- $(GU_n,SU_n)$ if $p$ is coprime to $n$ and $n$ is odd.

One can also show that multiplicity $1$ holds for certain kinds of representations, rather than for all representations of certain groups. For example, it easily holds for generic representations, due to the uniqueness of Whittaker models. Monica Nevins has shown that multiplicity one always holds for certain kinds of supercuspidal representations. More generally, Manish Mishra and I can show that multiplicity one holds for Kaletha's ``regular'' supercuspidal representations that satisfy an additional regularity condition, which is equivalent to the components of the restricted representation remaining regular.

Some or all of the examples above are predicted by the conjecture.

It has long been known that higher multiplicities can occur if $G= SL_1(D)$ for $D$ a division algebra (I apologize for not having the reference at hand), so one should expect similar behavior for many other groups involving division algebras. See one of the papers above for an example of higher multiplicity that can occur for the pair $(GU_n, SU_n)$, even when our unitary groups are quasi-split. The example involves a regular supercuspidal representation whose restriction breaks up into components that are not regular.