There is a concrete but disappointing answer to your question. Let me discard $C_c(E)$ as it is not a Banach space and let us focus on what you call $C_\infty$ but it is more commonly denoted by $C_0(E)$. In this case the dual space is the space of all Borel measures on $E$ with the total variation norm, see also this thread.

The space $C_b(E)$ can be humongous. For instance, when $E = \mathbb N$ it is naturally identifiable with $\ell_\infty$, the space of all bounded sequences. More generally, you may identify that $C_b(E)$ is canonically isometrically isomorphic to $C(\beta E)$, where $\beta E$ stands for the Čech–Stone compactification of $E$. So by the Riesz representation theorem, the dual of $C_b(E)$ is just the space of Borel measures on $\beta E$. You probably can't do better as already for $E=\mathbb N$ this dual space is a mess.

There is a concrete but disappointing answer to your question. Let me discard $C_c(E)$ as it is not a Banach space and let us focus on what you call $C_\infty$ but it is more commonly denoted by $C_0(E)$. In this case the dual space is the space of all Borel measures on $E$ with the total variation norm, see also A question on the Riesz-Markov theorem about dual space of $C_0(X)$.

The space $C_b(E)$ can be humongous. For instance, when $E = \mathbb N$ it is naturally identifiable with $\ell_\infty$, the space of all bounded sequences. More generally, you may identify that $C_b(E)$ is canonically isometrically isomorphic to $C(\beta E)$, where $\beta E$ stands for the Čech–Stone compactification of $E$. So by the Riesz representation theorem, the dual of $C_b(E)$ is just the space of Borel measures on $\beta E$. You probably can't do better as already for $E=\mathbb N$ this dual space is a mess.