I have some difficulties to see why the definition of the lie derivative makes sense.
I'm going to use the notations of Lee's book.
Let $M$ manifold and $W$ a smooth vector field on $M$. Let $\eta$ the flow of another smooth vector field $V$
$$(L_VW)_p=d/dt(d(\eta_{-t})_{\eta_t(p)}(W_{\eta_t(p)})=\lim_t (d(\eta_{-t})_{\eta_t(p)}(W_{\eta_t(p)})-W_p)/t $$
where the derivative and the limit are computed in $0$
Now:
1) Is the second equality a definition ?
2) I don't understand what a derivative or a limit of these objects means. I would like to find a function $F$ depending on $t$ such that $(L_VW)_p=d/dt F(t)$ computed in $0$. My problem is that if I take $F(t)=d(\eta_{-t})_{\eta_t(p)}(W_{\eta_t(p)})$ I have a function with values not in $\mathbb R $ (I think they are in $T_pM$, right?) So how can I derive a such $F:\mathbb R\to T_pM$ ?
Is this problem just that I do not know what a derivative in a topological vector space is or something like this?
Same problem with the Lie derivative of a tensor field.
P.S. Moreover, the symbol $d/dt$ in the definition of the Lie derivative should be interpreted as the velocity of a curve or as the limit of the difference quotient? Or, more in general, the velocity of a curve into $T_pM$ is actually equals to the limit of the different quotient?
It is not as bad as you fear: $F(t)=d(\eta_{-t})_{\eta_t(p)}(W_{\eta_t(p)})$ takes, for any value of $t$, value in $T_p M$. You surely can take the difference of two vectors, $F(t)$ and $F(0)$, which belong to the same vector space. Similarly you can divide by a constant ($t$) and take limits. If you don't like thinking of limits in $T_p M$ just remember that $T_p M \simeq \mathbb{R}^n$ for $n=$dim$(M)$. You probably have seen derivatives and limits of function valued in $\mathbb{R}^n$, it is just that. To answer (1), the second equality is not a definition, but follows from the definition of derivative by what I said before.