You only really need to understand one example to understand the case of any finite group: $\widehat{\mathbb{Z}/n\mathbb{Z}}$.
Indeed, it's not hard to show that $\widehat{A\times B}\cong \widehat{A}\times\widehat{B}$ by the obvious maps (this is just saying that Hom is additive in the first entry). From this and the structure theorem we see that $\widehat{A}$ is just a product of groups of the form $\widehat{\mathbb{Z}/n\mathbb{Z}}$.
Now, if $G$ is a non-abelian group we merely note that
$$\text{Hom}_\mathbf{Grp}(G,\mathbb{C}^\times)=\text{Hom}_\mathbf{Grp}(G^\text{ab},\mathbb{C}^\times)$$
where $G^\text{ab}$ is the abelianization of $G$ (i.e. $G/[G,G]$). Thus, we see that
$$\widehat{G}\cong\widehat{G^{\text{ab}}}$$
So, we really can restrict ourselves to looking at $\widehat{\mathbb{Z}/n\mathbb{Z}}$.
Well, what is a homomorphism $\mathbb{Z}/n\mathbb{Z}\to\mathbb{C}^\times$? It's just specifying an element $z$ of $\mathbb{C}^\times$ for which $1$ maps to, and the only condition on $z$ is that $z^n=1$. Thus, we are really looking at mappings $\mathbb{Z}/n\mathbb{Z}\to \mu_n$ (where the latter is the group of $n^{\text{th}}$ roots of unity). But, it's clear that $1$ can map to any element of $\mu_n$, and thus we see that specifying $\mathbb{Z}/n\mathbb{Z}\to\mu_n$ is really just picking some $e^{\frac{2\pi i r}{n}}$ for which $1$ should map to.
In fact, it's not hard to see that from this that the mapping
$$\mathbb{Z}/n\mathbb{Z}\to\widehat{\mathbb{Z}/n\mathbb{Z}}:r\mapsto \left(1\mapsto e^{\frac{2\pi i r}{n}}\right)$$
is actually an isomorphism of groups!
Thus, as groups, $\widehat{\mathbb{Z}/n\mathbb{Z}}$ is nothing but $\mathbb{Z}/n\mathbb{Z}$, and the isomorphism is very explicit. Following previous statements you can then deduce that $\widehat{G}\cong G^\text{ab}$, and the isomorphism is fairly explicit. In fact, you should prove to yourself that if $\widehat{\mathbb{Z}}:=\text{Hom}(\mathbb{Z},\mathbb{T})$, then $\widehat{\mathbb{Z}}\cong\mathbb{T}$(where $\mathbb{T}$ is the circle group), and so you can extend the above calculation to finitely generated groups $G$.
The above is actually just the very beginning of a rich and beautiful subject. Namely, we can extend the above notion of a dual group from a finite group, or a finitely generated group, to any locally compact abelian group. The correct notion of dual there is the Pontryagin Dual of a locally compact group $G$, which is defined to be the group of continuous homomorphisms $G\to\mathbb{T}$ with the obvious group structure, and with the topology of uniform convergence on compact subsets. There it is not true that $G\cong\widehat{G}$ (we have seen one example of this). But, there is the celebrated Pontryagin duality theorem which states that $\widehat{\widehat{G}}\cong G$ in a natural way for all $G$.
This level of generality is the most illuminating way to understand many of the common notions in mathematics, such as the Fourier transform. In fact, it is ubiquitous in modern mathematics, from analysis (e.g. the Fourier transform) to number theory (the character group of various local and global fields).
Best Answer
Every element of $\mathbb{Q}/\mathbb{Z}$ can be written uniquely in the form $q+\mathbb{Z}$ for $q\in [0,1)$ (if this isn't obvious, you should take a moment to prove it). So, let $G=\{0,g_1,g_2,\cdots,g_n\}$ be a finite group, and denote $n_i:=\vert g_i \vert$. Then, for all $f\in G^*$, $n_i f(g_i)=0+\mathbb{Z}$. Write $f(g_i)=\frac{a_i}{b_i} + \mathbb{Z}$ with $a_i,b_i\in \mathbb{N}$, $a_i<b_i$, and $gcd(a_i,b_i)=1$. Then, in $\mathbb{Z}$, we have $b_i \vert a_i n_i$ which implies $b_i \vert n_i$. Since there are only finitely many positive integer divisors of $n_i$ and only finitely many corresponding choices for $a_i$, there are a finite number of ways to choose $f$. That takes care of the first question.
For the second, I'll bet you can show that the map sending 1 to $\frac{1}{n}+\mathbb{Z}$ is a generator for $G^*$ by using an argument similar to the one above.