If $\lambda$ has at most two rows then the multiplicity of $S_\lambda$ is given by the Cayley–Sylvester formula. This gives the complete decomposition if $\dim V = 2$. If $\lambda$ is of `near-hook' shape then there are relevant results in this paper by Giannelli.

The unique maximal constituent is $S_{(3k)}$. For any $m$, the minimal partitions $\lambda$ in the dominance order, such that $S_\lambda$ appears in $\mathrm{Sym}^k(\mathrm{Sym}^m(V))$ are classified in a joint paper with Rowena Paget. (For $m$ odd we later learned that this result follows from earlier work of Klivans and Reiner.) To give a flavour of these results, when $k=m=3$, the minimal partitions are $(5,2,2)$ and $(4,4,1)$; the first corresponds to the set family $\{ \{1,2,3\},\{1,2,4\},\{1,2,5\} \}$ and the second to the set family $\{ \{1,2,3\}, \{1,2,4\}, \{1,3,4\} \}$. In the Klivans–Reiner setting, these become highest weight vectors in $\bigwedge^3\bigl(\bigwedge^3 V\bigr)$. For instance, if $V = \langle e_1, e_2, \ldots \rangle$ then the first corresponds to

$$(e_1 \wedge e_2 \wedge e_3) \wedge (e_1 \wedge e_2 \wedge e_4) \wedge (e_1 \wedge e_2 \wedge e_5) \in \bigwedge^3\Bigl( \bigwedge^3V \Bigr).$$

Another result that can be proved using highest weight vectors is
Proposition 4.3.4 in Ikenmeyer's thesis: if $S_\lambda$ appears in $\mathrm{Sym}^k(\mathrm{Sym}^3(V))$ and $S_\mu$ appears in $\mathrm{Sym}^\ell(\mathrm{Sym}^3(V))$ then $S_{\lambda + \mu}$ appears in $\mathrm{Sym}^{k+\ell}(\mathrm{Sym}^3(V))$.

The $S_\lambda$ appearing in $\mathrm{Sym}^3(\mathrm{Sym}^k(V))$ were found for all $k$ by Dent and Siemons, who showed they all appear, with at least the relevant multiplicity, in $\mathrm{Sym}^k(\mathrm{Sym}^3(V))$. This proves a special case of Foulkes' Conjecture.

For small $k$ the complete decomposition can be found using symmetric functions or other methods. There are data on my website for $k \le 15$, computed using Proposition 5.1 in this joint paper with Anton Evseev and Rowena Paget.

None of these results comes close to giving a complete picture. I have looked, without success, for a combinatorial rule that gives the constituents of $\mathrm{Sym}^{k+1}(\mathrm{Sym}^3(V))$ from $\mathrm{Sym}^{k}(\mathrm{Sym}^3(V))$, analogous to Young's Rule or Pieri's Rule.