Sternberg in his "Lectures on differential geometry" considers the set $Cu(M)$ of all pairs $(p,\gamma)$ with $p\in M$ and $\gamma:(-\epsilon,\epsilon)\to M$ differentiable with $\gamma(0)=p$. Evidently, $Cu(M)$ is the space of all curves in $M$. Now on $Cu(M)$ he introduces an equivalence relation $$ (p,\gamma) \sim (\tilde{p},\tilde{\gamma}) \ :\Leftrightarrow \ p=\tilde{p} \ \text{and} \ \ \left.\dfrac{dx^\mu \circ \gamma}{dt}\right|_{t=0}=\left.\dfrac{dx^\mu \circ \tilde{\gamma}}{dt}\right|_{t=0}. $$ It is $Cu(M)/_\sim$ that is what is called the tangent bundle $TM$.
He argues then that he needs to exhibit that the equivalence relation is independent of the chart $x:U\subset M\to \mathbb{R}^n$. This is where my question lies:
Much like when I have a map $f:A/_\sim \to B$, where I use a representative to define the map and then I have to show that the map is well-defined by showing that the image point is independent of the representative chosen to define the map; it seems to me that choosing a chart is similar to choosing a representative, and whenever I define something in terms of a chart, I need to make sure that what I define is independent of the representative. So, in the case of a smooth manifold, what is the coset space from which I pick charts?
I don't really understand what do you mean but maybe what your intuition tells you is this.
Usually, there are three different kind of construction of tangent space $T_p M$ of a manifold.
The first, which is the most geometric and intuitive way is by defining it as equivalence class of curves. That is if $p\in M$, we define the elements of $T_pM$, the tangent vectors as equivalence class of curves $[c]$ that pass through $p$. The equivalence relation define by $c_1 \sim c_2$ where $c_1(0)=c_2(0)=p$ if and only if $(f \circ c_1)'(0) = (f \circ c_2)'(0)$ as you reference define it. This kind of tangent space is $T_pM = \scr{C}_p/\sim$ where $\scr{C}_p$ is set of all curves defined in some small interval contain $0$ that pass $p \in M$.
The second formulation is an abstract one rely heavily on the concept of derivation. In this case, the tangent vector $v_p$ defined as a linear map $v_p : C^{\infty}(M) \rightarrow \mathbb{R}$ which is obeys product rule $$ v_p(fg) = f(p) v_p g + g(p) v_p f $$ This is the most common one and easy to work with despite the lack of intuitiveness.
The third (i think this is that you're asking) is by defining $T_pM$ via equivalence class of charts. For a fix $p\in M$, we consider the set $\Gamma_p$ consists of all triplets $(p, v, (U,x)) \in \{p\} \times \mathbb{R}^n \times \scr{A}$ such that $p\in U$ ($\scr{A}$ is the maximal atlas for $M$.). And define an equivalence relation on $\Gamma_p$ by requiring that $((p, v, (U,x)) \sim (p, w, (V,y)))$ if and only if $$ w = D(y \circ x^{-1})|_{x(p)} \cdot v $$ That is we define tangen vector as an object that doesnt change under coordinate transformation. The equivalence class of this $[(p, v, (U,x))]$ is defined as tangent vector in $T_pM$. So $T_pM = \Gamma_p / \sim$.
All of this formulation is actually same. That is we can identified one with another by isomorphism (Lookmore details in Jeffrey Lee's book or Conlon's book).