You seem want to introduce gauge invariance into a theory that doesn't appear to have the global symmetry need in the first place. One way to think of gauge invariance is that you 'gauge' the global symmetry, then you just change your derivative terms to covariant derivatives like you mentioned. In other words, we can only concern ourselves with the global symmetry for now and gauge it at the very end if we wish. Now, at the very least in the term you wrote down
$$\phi \bar{\psi} \psi$$
it's not clear how the indices are contracted. Do the $\phi$ and $\psi$ have indices? For example I could make the $\psi$ transform in under $SU(2)$ by introducing a second copy of the $\psi$ and sum over them:
$$ \phi \bar{\psi^a} \psi^a $$
but now I can't make the $\phi $ transform because there isn't anything left to contract the $\phi$ index with. That is,
$$ \phi^b \bar{\psi^a} \psi^a $$
isn't a singlet (a singlet doesn't transform under the symmetry ) so it doesn't make any sense as a term in your lagrangian. Or you could introduce a second spinor that is a singlet under the global symmetry so you have something to contract your scalar indices with:
$$ \phi^a \bar{\psi^a} \eta $$
Finally, if you want a spinor to transform as a triplet, or in the adjoint of $SU(2)$ you could introduce the generator of SU(2) and contract indices as follows:
$$ \phi^a (t^a)^{bc} \bar{\psi^b} \psi^c = \phi^a \bar{\psi} t^a \psi$$
where the $t^a$ is in the fundamental (or doublet) representation so that we can properly trade the adjoint index for two doublet indices and get an overall singlet for the lagrangian.
Now, as for the kinetic terms, like you mentioned, if you want to introduce gauge symmetry trade the regular derivative for a covariant derivative.
$$ \partial_\mu \phi^a \rightarrow D_\mu \phi^a =\partial_\mu \phi^a + i {A_\mu}^b (t^b)^{ac}\phi^c $$
where the $t^a$ is in whatever rep the $\phi^a$ transforms in. The same holds for the fermion fields.
There is one caveat to all this however - this prescription of naively gauging a global symmetry that I have outlined breaks down if the global symmetry is 'anomalous'. That is, quantum mechanical effects break the naive, classical global symmetry. I'm not going to get into what that is, but keep it in the back of your mind for the time being and read about it when you have a chance.
I have a feeling you might want more info than this, but I'll stop here for now and if you edit I will clarify/ add.
EDIT: In retrospect this seems to work more easily for $SU(2)$ reps easier than other groups since for $SU(2)$ the adjoint rep is the same as the triplet rep so I can trade triplet rep indices for doublet indices using the generators $(t^a)^{ij}$. I am not sure if you can do these sort of things for groups in general.
Yes, parity is really violated, even if neutrinos are massive. You seem to be confusing the relationship between parity, helicity, and chirality in the modern standard model with the physical symmetry operation of spatial inversion.
Wu's experiment did not measure neutrino helicity. Wu and collaborators prepared a thin layer of a beta-emitting nucleus with rather high spin, polarized the nuclei, and observed that the beta particles were more likely to be emitted from the "south pole" of the nucleus than from the "north pole." Why is this a parity violation? The way you define the "north pole" of a rotating object is to grab it with your right hand, so that the fingers of your right hand curl around in the same sense as the rotation; the "north pole" is the one where your thumb goes, and the "south pole" is the other one. Mirror reflection (which is a special case of the parity transformation) turns your right hand into a left hand, and changes which pole you label as "north" for the same sense of rotation.
The following paper in that issue of PRL, by Garwin et al, shows that muons produced in the decay $\pi\to\mu+\nu$ are polarized. Such polarization is a violation of parity symmetry because the pion has zero spin, and a stopped pion can therefore express no preference about the spin directions of its daughters. This experiment was also the first to measure the magnetic moments for the muon and antimuon.
The statement that one can produce polarized $\mu$ from spinless $\pi$ in the decay $\pi\to\mu+\nu$ suggests that in that decay the neutrinos should also be produced polarized. However the first measurement of neutrino helicity is generally taken to be the one by Goldhaber and collaborators, later in the same year. Analysis of the Goldhaber experiment requires you to spend a little time thinking carefully about nuclear spins.
It was actually discovered in 1927 that unpolarized beta sources produce slightly left-handed polarized electrons, though the significance was not understood at the time and the paper was forgotten until unearthed by Grodzins in 1958. Allan Franklin calls it "the non-discovery of parity non-conservation."
Parity violation is real in the visceral sense that if you were to show me a (sufficiently detailed) photograph of a weak interaction experiment with polarized particles, I could in principle tell you whether the photograph had been reflected or not.
Best Answer
Groups are abstract. They have elements that can be "multiplied" and they have other properties (for example, every element has an inverse, etc). Representations of groups are concrete. For example a 3x3 matrix could be used to represent an abstract group element. Representations of groups can be multiplied, for example matrices can be multiplied by matrix multiplications.
Say, for example, that you have three abstract elements of a group called "a", "b", and "c". And say that $$ a\cdot b=c $$
Say, also, that you have a specific representation of those three above-mentioned group elements, the representation being 3x3 matrices called: R(a), R(b), and R(c). Then, you want the same multiplication rules to hold for the representation as for the group itself, namely: $$ R(a)\cdot R(b)=R(c)\;, $$ where the $\cdot$ now means matrix multiplication.
For every element in the group there should be some element in the representation of the group, and all the multiplication rules that hold for the elements of the group should hold for the elements of the representation.
The representation of the group does not have to be 3x3 matrices. There are an infinite number of different representations of a group.
One interesting representation is 1. I.e., the representation in which every element of the group is represented by 1. This is trivial, right? Yeah, that's why it is called the "trivial representation". But it works because, suppose that $a \to 1$, $b \to 1$, and $c \to 1$. Then $$ a\cdot b = c \to 1\cdot 1 = 1 $$ which is true.
So every group has a trivial representation.
When we say something "transforms under" some representation of the group, we mean that it gets multiplied by some representation of the group under the group transformation. If something transforms as a singlet it is transforming under the trivial representation. I.e., it gets multiplied by 1. I.e., it doesn't transform.
On the other hand, objects transforming under the fundamental representation of SU(3) get multiplied by 3x3 special unitary matrices. I.e., objects transforming under this representation are like 3x1 column vectors. I.e., triplets. There seems to be a commentator on your question that is confusing this issue.
There are other representations beside the trivial and the fundamental. As I said earlier there are an infinite number of representations. In the case of SU(3) there are also an infinite number of irreducible representations... but a discussion of this point is beyond the scope of this answer.