quantum field theory - Yukawa Coupling of a Scalar $SU(2)$ Triplet to a Left-Handed Fermionic $SU(2)$ Doublet

Suppose we have a field theory with a single complex scalar field $\phi$ and a single Dirac Fermion $\psi$, both massless. Let us write $\psi _L=\frac{1}{2}(1-\gamma ^5)\psi$. Then, the Yukawa coupling of the scalar field to the left-handed Fermion field should be of the form

$$g\overline{\psi}\psi \phi,$$ where $g$ is the coupling constant. So far, so good (at least I think; please correct me if this is incorrect).

Now, we introduce gauge invariance into the theory and demand that $\phi$ transform as a triplet and $\psi _L$ transform as a doublet under the gauge group $SU(2)$. What form does the Lagrangian take now? My confusion is arising because now, in particular, although $\phi$ transforms as a scalar under the Lorentz group, it must be described by $3$-components so that it can transform as a triplet under $SU(2)$. But then, the Yukawa coupling term listed above, as written, is not a number! I know this has something to do with the fact (I think) that $\overline{2}\otimes 2$ decomposes into something involving the triplet representation of $SU(2)$. Unfortunately, I don't know enough about the representation theory of $SU(2)$ to turn this into a Yukawa coupling term that makes sense.

Once again, because of my lack of knowledge of the representation theory of $SU(2)$, I don't know how to write the gauge covariant derivative corresponding to the triplet representation of $SU(2)$. If we use the Pauli matrices as a basis for $\mathfrak{su}(2)$, how is the triplet representation of $SU(2)$ described in terms of the Pauli matrices acting on a three-dimensional complex vector space?

I am also unsure as to exactly what should happen to the kinetic term for the Fermion field. Before insisting upon gauge invariance, this term should be of the form

$$i\overline{\psi}\gamma ^\mu \partial _\mu \psi .$$ However, because (I believe) gauge invariance is only being demanded for $\psi _L$, presumably something more complex happens to this term than just $i\overline{\psi}\gamma ^\mu D_\mu \psi$, where $D_\mu$ is the appropriate gauge covariant derivative. Instead, should this kinetic term be written in the form $$i\overline{\psi_L}\gamma ^\mu D_\mu \psi _L+i\overline{\psi _R}\gamma ^\mu \partial _\mu \psi _R?$$