If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like $c_1(L)\in H^2(X,\textbf Z)$.
But if $X$ is a scheme (say of finite type) over any field, then I saw a definition of the first Chern class $c_1(L)$ just via its action on the Chow group of $X$, namely, on cycles it works as follows: for a $k$-dimensional subvariety $V\subset X$ one defines
\begin{equation} c_1(L)\cap [V]=[C], \end{equation}
where $L|_V\cong\mathscr O_V(C)$, and $[C]\in A_{k-1}X$ denotes the Weil divisor associated to the Cartier divisor $C\in\textrm{Div}\,V$ (the latter being defined up to linear equivalence). So then one shows that this descends to rational equivalence and we end up with a morphism $c_1(L)\cap -:A_kX\to A_{k-1}X$. So, my naive questions are:
$\textbf{1.}$ Where do Chern classes "live"? (I just saw them defined via their action on $A_\ast X$ so the only thing I can guess is that $c_1(L)\in \textrm{End}\,A_\ast X$ but does that make sense?)
$\textbf{2.}$ How to recover the complex definition by using the general one that I gave?
$\textbf{3.}$ Are there any references where to learn about Chern classes from the very beginning, possibly with the aid of concrete examples?
Thank you!
First let me lift the suspense: if $L$ is a line bundle and if your scheme $X$ has dimension $n$, then $c_1(L)\in A^1(X)=A_{n-1}(X)$, where $A(X)$ is the Chow group of $X$, graded by codimension (upper indices) or dimension (lower indices).
The definition is very simple: take a non-zero rational section $s\in \Gamma_{rat}(X,L)$.
Its divisor of zeros and poles $div(s)$ is a cycle of dimension $n-1$ and the rational equivalence class of that cycle is the requested first Chern class: $$c_1(L)=[div(s)]\in A_{n-1}(X)$$
If $X$ is smooth (or if you want to be more technical, just locally factorial) the first chern class yields an isomorphism $$c_1: Pic(X)\xrightarrow \cong A^1(X)\quad (*)$$
If $X$ is a smooth variety defined over $\mathbb C$, then $A(X)$ has the structure of a ring graded by codimension and there is a canonical morphism of graded rings $A^*(X)\to H^{2*}(X^{an},\mathbb Z)$, sending $c_1(L)\in A^1(X)$ to the analytically defined Chern class $c_1(L^{an})\in H^2(X^{an},\mathbb Z)$ obtained by the exponential sequence.
(More generally $A^i(X)$ is sent to $H^{2i}(X^{an},\mathbb Z)$: that's what the notation with the stars above means)
The canonical (but very difficult) reference is of course Fulton's Intersection Theory.
Edit
A more accessible resource is a projected book by Eisenbud and Harris , amusingly called 3264 & All That, a draft of which they generously put online.
Second Edit
As an answer to a question in atricolf's comment, note that the displayed isomorphism $(*)$ implies that in general $A^1(X)$ is very far from being isomorphic to $H^{2}(X^{an},\mathbb Z)$.
For an elliptic curve $X$, for example, $A^1(X)$ is isomorphic to $X\times \mathbb Z$, which has the cardinality $2^{\aleph_0}$ of the continuum, whereas $H^{2}(X^{an},\mathbb Z)$ is isomorphic to $ \mathbb Z$.