I am new to differential geometry and I would appreciate it if someone could help me answering these questions.
1- Why does the set of all smooth diffeomorphisms of the circle $S^1$ form a smooth manifold? I assume we need to construct local charts but the details are not so obvious to me.
2- Main question: to find a tangent vector to $\text{Diff}(S^1)$ we take a curve $\xi_t$ in $S^1$ (sequence of transformations with parameter $t$), then we have:
$\dot{\xi_t}(x) = \frac{d}{dt} \xi_t(x) = \lim_{\tau \to 0} \frac{\xi_{\tau}(x) - 1}{\tau} \xi_t = u(x)\circ\xi_t(x)$
and in this book that I am reading it is mentioned that "So any tangent vector of $T_x\text{Diff}(S^1)$ is represented as $X=u(x)\partial_x$". Does the operator $\circ$ in the equation above mean normal function multiplication? And how can one deduct the form $X=u(x)\partial_x$ directly from the limit definition?
Thank you very much for the help.
To partly answer at least your second question, the formula you typed: $$ \dot{\xi_t}(x) = \frac{d}{dt} \xi_t(x) = \lim_{\tau \to 0} \frac{\xi_{\tau}(x) - 1}{\tau} \xi_t = u(x)\circ\xi_t(x) $$ seems to be wrong, in the sense that the types don't match for the second equality. I think it should be something akin to $$ \dot{\xi_t}(x) = \frac{d}{dt} \xi_t(x) = \lim_{\tau \to 0} \frac{\xi_{\tau}(x) - 1}{\tau} \xi_t(x) = u(x)\circ\xi_t(x) $$ although this requires that $$ \xi_a(x) \xi_b(x) =\xi_{a+b}(x) $$ which means that the path $t \mapsto \xi_t$ has a kind of 'homomorphism' structure to it that doesn't seem to be assumed in what you've said.
It also seems to require that $\xi_\tau(x)$ be a real (or possibly complex) number, since $1$ gets subtracted from it, i.e., it assumes that $S^1$ is being treated here as the set of complex numbers of modulus 1. Alternatively, it could be that $1$ here denotes the identity function, and the proper form is $$ \dot{\xi_t}(x) = \frac{d}{dt} \xi_t(x) = \lim_{\tau \to 0} \frac{\xi_{\tau} - 1}{\tau} (\xi_t(x)) = u(x)\circ\xi_t(x) $$ so that the limit is a limit of functions, and these functions are applied to the point $\xi_t(x)$. But that doesn't seem to make sense for other reasons, like "Why should $\xi_\tau(\xi_t(x))= \xi_{t + \tau}(x)$?" That kind of statement does hold true for things like $$ c_a(x) = q(x + a) $$ where $q$ is a path, and $c_a$ is the path "offset by time $a$", which often occur in "one-parameter families of geodesics", for instance...but I see nothing in what you've told us that suggests that it holds here.
I suspect that the author was trying to make the point that the thing inside the limit depends only on $x$, not $t$, but as for the meaning of the little circle, or any explanation of why this sequence of equalities is correct...I'm at a loss.
I've actually taught differential geometry two or three times in my career, perhaps not very well, and only at the undergraduate level. I took courses in it from both Singer and Chern, which gave me views of the subject from two rather different sides of things, back when I was at Berkeley. And I can't make sense of this stuff as written. Perhaps you're just a great deal more insightful than I am, and will make sense of it. But it's hard to believe that you'll do so faster than you can read the two other books I suggested, each of which will lead you to actually understand the basics of the subject.
Seriously: I recommend that you find a book that's more clearly written.