I am confused with the idea of tangent vector or tangent space. First of all, I learned that there is an isomorphism from $ \mathbb{R}_a^n$ onto $T_a( \mathbb{R} ^n)$ from John M.Lee' book Introduction to Smooth Manifolds. Although we have the perspective of regarding tangent vectors as an operator defined on $\mathbb{R} ^n$ or more generally, a manifolds, I am still have trouble with it. Again, on Lee's book,
For example, any geomantic tangent vector $v_a \in \mathbb{R} _a^n$ yields a map $D_{v,a}:C^\infty ( \mathbb{R} ^n)\to \mathbb{R} $, which takes the directional derivative in the direction $v$ at $a$: $$D_{v,a}f = D_v f(a) = \frac{d}{dt} f(a+tv)$$
Here are my questions: Now considering a special manifold, a surface embedded in $\mathbb{R}^3$ and the tangent of the surface. I know we have to define a smooth function $f$ on the (special) manifold and we must define a function $f$ with three dimensions in order to take directional derivative by a three dimensional vector $v_a$. However, our manifold is a two dimensional surface embedded in $\mathbb{R} ^3$. Unfortunately, $2\neq 3$. So, what's dimension of the domain of $f$ with respect to the special manifold?
Edit:@Jack Lee: He points out that tangent vectors to the sphere are defined more abstractly as derivations. At first we have Euclidean space,$\mathbb{R}^3$, and the space $\mathbb{R}_a^n$, so we have its tangent space $T_p(\mathbb{R}^3)$ and we prove that these two linear vector space are isomorphic. We then define the general tangent space w.r.t manifolds.
Let $M$ be a smooth manifold with or without boundary, and let $p$ be a point of $M$. A linear map $v:C^\infty(M)\to R$ is called a derivation at $p$ if it satisfies $$v(fg) = f(p)vg + g(p)vg$$ for all $f,g \in C^\infty(M)$. The set of all derivation of $C^\infty(M)$ at $p$, denoted by $T_p M$, is a vector space called the tangent space to $M$ at $P$. An element of $T_p M$ is called a tangent vector at $p$.
The author then discuss the sub-manifolds and the relation between the tangent space of the embedding space and its ambient space.
Let $M$ be a smooth manifold with or without boundary, and let $S\subseteq M$ be an immersed or embedded submanifold. Since the inclusion map $\iota:\hookrightarrow M$ is a smooth immersion, at each point $p\in S$ we have in injective linear map $d_{\iota_p}:T_p S\to T_p M$. In terms of derivations, this injection works in the following way: for any vector $v\in T_p S$, the pimage vector $\tilde{v} = d_{\iota_p}(v)\in T_p M$ acts on smooth functions on $M$ by $$\tilde{v}f = d_{\iota_p}(v)f = v(f\cdot\iota) = v(f\big|_S)$$
Finally, there is also a picture illustrated on the book help a lot.
I think you're trying to read more into this definition than is there. The definition you quoted is only talking about tangent vectors to $\mathbb R^n$, not to submanifolds of $\mathbb R^n$ such as the sphere. Tangent vectors to the sphere are defined more abstractly as derivations (see p. 54). The relationship between tangent vectors to $\mathbb R^n$ and tangent vectors to a submanifold like the sphere isn't developed until Chapter 5.