I'm trying to understand the definition the adelic integral given in Goldfeld and Hundley. It says that:
Suppose that $f =\prod_v f_v$ is a factorizable function, that $f_\infty$ is an integrable function on $\mathbb{R}$, that for each $p$, the function $f_p$ is the characteristic function of a compact set $C_p$, and that $C_p={\mathbb{Z}_p}$ for all $p$ outside of some finite set $S$. Then we define the adelic integral $$ \int_{\mathbb{A}_\mathbb{Q}}f(x) dx = \int_{\mathbb{R}} f_\infty(x_\infty) dx_\infty \cdot \prod_{p\in S}\int_{\mathbb{Q}_p}f_p(x_p)dx_p. $$
Here $dx$ is wrt. the Haar measure. I have trouble understanding how we determine $S$ and $C_p$. I would like to calculate the specific example where $f=1_{\mathbb{A}_\mathbb{Q}}$ or when $f(x)=e(x):=\prod_{v\leq \infty} e_v(x_v)$ and $e_v(x_v):= e^{-2\pi i \{x\}}$. In both cases $f$ is factorizable, so one can in fact take the adelic integral. If $f=1$ then $$ \int_{\mathbb{A}_\mathbb{Q}}1 dx = \int_{\mathbb{R}} 1 dx_\infty \cdot \prod_{p\in S}\int_{\mathbb{Q}_p}1 dx_p $$ but what is $S$?
As KCd already remarked in the comments, the functions you came up with are not examples since they do not belong to the class of functions described by the authors. For instance, the constant function $f_{\infty}=1$ is not integrable on $\mathbb{R}$ and your function $f=1_{\mathbb{A}}$ is therefore not an example. The function $e_{\infty}(x)=e^{-2\pi i x}$ is not integrable on $\mathbb{R}$ and thus, $e=\prod_v e_v$ is not an example. Also note that your functions $f$ and $e$ do not satisfy the required conditions for the finite places. For example, if $p$ is a prime number, the constant function $f_p=1$ is not the characteristic function of a compact set in $\mathbb{Q}_p$.
Here is a recipe for making examples for integrable functions on $\mathbb{A}$. This should also clarify what the sets $S$ and $C_p$ are.
Step 1: For $f_{\infty}$, choose any integrable function on $\mathbb{R}$ you like. For example, you can take the characteristic function of a finite interval $[a,b]$. Another example is $f_{\infty}(x)=e^{-x^2}$.
Step 2: For $S$, choose any finite set of prime numbers. $S=\emptyset$ is also allowed.
Step 3: For each $p\in S$, choose any compact set $C_p\subseteq\mathbb{Q}_p$ you like. Typical examples for $C_p$ are fractional ideals in $\mathbb{Q}_p$.
Step 4: For each prime $p$, define the function $f_p:\mathbb{Q}_p\to\mathbb{R}$ as follows: $$f_p=\begin{cases} 1_{C_p} & \text{if }p\in S \\ 1_{\mathbb{Z}_p} & \text{if } p\notin S, \end{cases}$$ where $1_{C_p}$ and $1_{\mathbb{Z}_p}$ are the characteristic functions of the sets $C_p$ and $\mathbb{Z}_p$ respectively.
Step 5: An integrable adelic function $f:\mathbb{A}\to\mathbb{C}$ can then be defined by $$f(x):=\prod_v f_v(x_v) = f_{\infty}(x_{\infty}) \prod_p f_p(x_p)\text{.}$$ The product $\prod_p$ is taken over all primes $p$. Considering step 4, we can write $f$ more explicitly as $$f(x)=f_{\infty}(x_{\infty})\prod_{p\in S} 1_{C_p}(x_p)\prod_{p \notin S} 1_{\mathbb{Z}_p}(x_p).$$ The integral of $f$ over $\mathbb{A}$ is then given by \begin{align*} \int_{\mathbb{A}} f(x)dx &:= \int_{\mathbb{R}} f_{\infty}(x_{\infty})dx_{\infty} \; \prod_{p\in S}\int_{\mathbb{Q}_p}f_p(x_p)dx_p \\[5pt] &=\int_{\mathbb{R}} f_{\infty}(x_{\infty})dx_{\infty} \; \prod_{p\in S}\int_{\mathbb{Q}_p}1_{C_p}(x_p)dx_p\text{.} \end{align*} So, the set $S$ is the one we have chosen in step 2.
Below I will state a more concrete example. Before you go through it, make sure you understand why the functions constructed according to the above steps are indeed of the class described in your textbook and that all functions from that class are constructed in this way. (In fact, what I wrote above is exactly the definition of your textbook just written down in a more fine-grained way.)
Example: Take $f_{\infty}(x)=e^{-x^2}$ for $x \in \mathbb{R}$. We choose the set $S$ to be $\{2,3\}$. Now we define $f_p$ at the primes $p=2$ and $p=3$: Let $\mathfrak{a}$ be some fractional ideal in $\mathbb{Q}_2$ and $\mathfrak{b}$ be some fractional ideal in $\mathbb{Q}_3$. Note that it is easy to describe $\mathfrak{a}$ and $\mathfrak{b}$ in more concrete terms: Every fractional ideal in $\mathbb{Q}_2$ is of the form $2^n\mathbb{Z}_2$ with some integer $n$. Similarly, the fractional ideals in $\mathbb{Q}_3$ are of the form $3^n\mathbb{Z}_3$. Now for $f_2$, we take the characteristic function of $\mathfrak{a}$ and for $f_3$, we take the characteristic function of $\mathfrak{b}$. For all primes $p>3$, we take $f_p$ to be the characteristic function of $\mathbb{Z}_p$. To summarize: $$f_p=\begin{cases}\text{char. function of }\mathfrak{a} & \text{if }p=2 \\ \text{char. function of }\mathfrak{b} & \text{if }p=3 \\ 1_{\mathbb{Z}_p} & \text{if } p\notin S. \end{cases}$$ As adelic function $f:\mathbb{A}\to\mathbb{C}$ we take $$f(x):=f_{\infty}(x_{\infty})\prod_p f_p(x_p).$$ Now we calculate the integral of $f$: \begin{align*} \int_{\mathbb{A}} f(x)dx &=\int_{\mathbb{R}} f_{\infty}(x_{\infty})dx_{\infty} \; \prod_{p\in S}\int_{\mathbb{Q}_p}f_p(x_p)dx_p \\[5pt] &=\int_{\mathbb{R}} e^{-x^2} dx \int_{\mathbb{Q}_2} f_2(x) dx \int_{\mathbb{Q}_3} f_3(x) dx \\[8pt] &=\sqrt{\pi}\; \mu_2(\mathfrak{a})\mu_3(\mathfrak{b})\text{,} \end{align*} where $\mu_2$ and $\mu_3$ are the Haar-measures on $\mathbb{Q}_2$ and $\mathbb{Q}_3$. The values of $\mu_2(\mathfrak{a})$ and $\mu_3(\mathfrak{b})$ can be calculated explicitly: If we write $\mathfrak{a}=2^n\mathbb{Z}_2$ and $\mathfrak{b}=3^m\mathbb{Z}_3$ with integers $n,m$, it can be shown that $$\mu_2(\mathfrak{a})=2^{-n},\phantom{aaa}\mu_3(\mathfrak{b})=3^{-m}\text{.}$$ Not sure if the calculation of Haar-measures of ideals is also covered in your textbook but I decided to mention it anyway.