Let $G=S_3$ and $H=\langle (12) \rangle < G$. Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$.
I have computed the character using the induction formula and believe the answer to be $(3,1,0)$.
Now in my notes I have that $\chi_{\mathbb{C}[G]}=(|G|,0,\dots,0)$. This is under a section called regular representation which is what I believe this is but this does not seem to make sense.
I believe I have a fundamental lack of understanding of something. There is also a probable link is this lack of understanding to this question Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order..
You are confusing the way $\mathbb{C}[G]$ and $\mathbb{C}[G/H]$ relate to the group $G$: The first refers to the regular represenation of the group $G$. That is the representation of $G$ via the way it permutes its own elements by (left) multiplication.
On the other hand $\mathbb{C}[G/H]$ is a representation of $G$ (not $G/H$ that might not even be a group...) via a different action; the action by left multiplication on the cosets of $H$.
In particular, in the two cases, $G$ acts on different sets; in the first it acts on $\{g: g\in G\}$ and in the second it acts on $\{gH :g\in G\}$.