I just recently started differential topology and I am really struggling with some of the ideas. In particular, once we start talking about tangent vectors. Let $M$ be a n-manifold, then a linear map $v:C^\infty(M)\to \mathbb{R}$ is called a derivation at $p$ if it satisfies the product rule so $\frac{d}{dx}$ for example is a derivation at $p$ in $\mathbb{R}$ right? Then we define the tangent space $T_pM$ to be the space of all derivations at $p$. Basically we take $v(f)=D_v|_p(f)$, the directional derivative at $p$ in the direction $v$. We have discussed a few properties and for the most part I think I understand. Where I get really confused is when we started talking about the differential.
Let $F:M\to N$ be smooth and $p\in M$. Then the differential $dF_p:T_pM\to T_{F(p)}N$ is defined to be $dF_p(v)(f)=v(f(F))$ for all $f\in C^\infty(N)$. This is confusing to me because we don't know what $dF_p(v)$ is. We only know how it acts on $f$. We have discussed some properties like how it is a derivation at $F(p)$, it is linear, and how the chain rule works. I didn't really follow a lot of this so if anyone would be willing to go through it slowly I would appreciate it.
Next we started discussing some computations of things, basically we talked about how $T_p\mathbb{R}^n=span\left\{\frac{\partial}{\partial x^1}|_p,...,\frac{\partial}{\partial x^n}|_p\right\}$ which makes sense to me. If our manifold $M$ has an atlas $\psi_p$ then $d\psi_p:T_pM\to T_{\psi(p)}\mathbb{R}^n$ is an isomorphism by locality and furthermore, we can compute a basis for $T_pM$ by $\frac{\partial}{\partial x^i}|_p=d\psi_p^{-1}(\frac{\partial}{\partial x^i}|_{\psi(p)})$.
Now what I didn't understand at all was the computations of $\frac{\partial}{\partial x^i}|_p f$ for $f\in C^\infty(U)$. Is this notation the same thing as $\frac{\partial f}{\partial x^i}|_p$ and how do we get from this to $\frac{\partial f(\psi^{-1})}{\partial x^i}|_{\psi(p)}$?
Next it was noted that every $v\in T_pM$ can be written uniquely as $\sum v^i\frac{\partial}{\partial x^i}|_p$ but what are these $v^i$? Next we computed $dF_p(v)$ in local coordinates and I didn't understand a single step because we only defined $dF_p(v)(f)$. Can someone explain how to do this?
I apologize for so many questions but I would really like to understand this material better. If possible can someone go through all of this slowly and give me a very basic example (maybe like $\mathbb{R}^3$ or $S^2$) of how I can see all of this.
Your function $F$ can be thought of as a function from $\mathbb{R}^n$ to $\mathbb{R}^m$, where $n$ and $m$ are the respective dimensions of the manifolds (just express them in local coordinates $(x_i)$ and $(y_j)$). Call this "down to earth" function $\tilde F:\mathbb{R}^n\to\mathbb{R}^m$ . Suppose at some point $a\in M$ with local coordinates $(a_1,a_2,\dots a_n)$, some tangent vector $v$ is given, for example $v=(1,0,0,\dots)$ so your "tangent vector" means derivation w.r.t. the first local coordinate $x^1$. Denote coordinates in $N$ by $y=(y_1,y_2,\dots y_m)$, that is $P\in N\to\psi(P)=y$. Then, given a function $f$ on $N$, you bring it "down" to $\mathbb{R}^m$, which means you consider $\tilde f:\mathbb{R}^m\to\mathbb{R}$ defined as $\tilde f(y)=f\circ \psi^{-1}(y)$. Now everything is happening between $\mathbb{R}^n$ and $\mathbb{R}^m$ and we simply compute the derivative of $\tilde f$ in the direction of $d\tilde F(a_1,a_2,\dots a_n)(v)$. The resulting number is $dF_a(v)(f)$.
If you consider this number over all possible $f$, you obtain a derivation (that is a tangent vector).
I think it is easier to think of tangent vectors as velocity vectors of curves going through the given point $a\in M$. Via the differential $dF$, those are transformed into tangent vectors to the image of the curve in $N$. This is a more "tangible" approach.