Here's a very special case for $\mathfrak{gl}_n$ in characteristic 0 (which I have found useful in my work). Let $V$ be the vector representation, and for a partition $\lambda$ with at most $n$ parts, let ${\bf S}\_{\lambda}(V)$ denote the corresponding highest weight representation. Then $Sym^n(Sym^2 V) = \bigoplus\_{\lambda} {\bf S}\_{\lambda}(V)$ where the direct sum is over all partitions $\lambda$ of size $2n$ with at most $n$ parts such that each part of $\lambda$ is even. Similarly, $Sym^n(\bigwedge^2 V) = \bigoplus\_{\mu} {\bf S}\_{\mu}(V)$ where the direct sum is over all partitions $\mu$ of $2n$ with at most $n$ parts such that each part of the conjugate partition $\mu'$ is even. If you want the corresponding result for $\mathfrak{sl}\_n$ we just introduce the equivalence relation $(\lambda\_1, \dots, \lambda\_n) \equiv (\lambda\_1 + r, \dots, \lambda\_n + r)$ where $r$ is an arbitrary integer.

One reference for this is Proposition 2.3.8 of Weyman's book *Cohomology of Vector Bundles and Syzygies* (note that $L\_\lambda E$ in that book means a highest weight representation with highest weight $\lambda'$ and not $\lambda$).

Another reference is Example I.8.6 of Macdonald's *Symmetric Functions and Hall Polynomials*, second edition, which proves the corresponding character formulas.