Suppose you have two objects, $ \phi _i $ and $ \psi _j $ that form representations of $ SU(2) $. With both fields in the fundamental representation, I believe there is one invariant, \begin{equation} \epsilon ^{ ij} \psi _i \phi _j \quad \end{equation}
But how would I extend the these ideas to the triplet representation for example? I know how to use Young tableaux to abstractly write down products of tensors in terms of their reducible representations, but I'm not sure how I can use that to form singlets in general.
As an example consider two doublets $\psi _i ,\phi_j $ and a triplet,$\Delta_k$ under $SU(2)$. Using Young tableaux I found, \begin{equation} ( {\mathbf{2}} \otimes {\mathbf{2}} ) \otimes {\mathbf{3}} = {\mathbf{5}} \oplus {\mathbf{3}} \oplus {\mathbf 1} \oplus {\mathbf 3} \end{equation} but I'm not sure how to isolate the triplet state to get a useful term (in physics language I want to put a $SU(2)$ singlet in my Lagrangian)
Although this will not answer your question, I can give two tips. First, you need to be careful with upper and lower indices. The equations you have written down are wrong because the indices do not match.
Second, you will need to use relations such as: \begin{equation} \frac{1}{2} \varepsilon_{ijk} \varepsilon^{klm} \varphi_{lm} = \frac{1}{2} (\delta_i^l \delta_j^m - \delta_i^m \delta_j^l) \varphi_{lm} = \varphi_{[ij]} \end{equation} to construct these tensors.