Can I take the vector space generated by {$1 , x, x^2$} over the field $F_3$? Let $V$ be the vector space over the field $F_3$ (field with $3$ elements) of dimension $3$. Define an action of the symmetric group $S_3$ on $V$ by $\sigma.e_i$= $e_{\sigma (i)}$ where  {$e_1$, $e_2$, $e_3$} is a basis for $V$. 
Can anyone please make me understand what this problem is saying?
I do not understand what is a vector space over the field $F_3$ (field with $3$ elements) ?
Can I  take the vector space generated by {$1 , x, x^2$} over the field $F_3$?
If the action is only on he basis elements then how would we find the operations like $ \sigma (e_1 + e_2)$?
 A: 
If the action is only on he basis elements then how would we find the operations like $σ(e_1+e_2)$?

This seems to be the key to your question.
You are right: at the outset, the action is explicitly only defined on the set $\{e_1, e_2, e_3\}$. Given a group element $\sigma$, and one of these basis elements $b$, we know what $\sigma\cdot b$ is, and that is all.
But every mapping defined on the basis of a vector space gives rise to a unique linear transformation extending that mapping to the entire vector space spanned by the elements. Furthermore, if the mapping is a bijection and the vector space is finite dimensional, the extension is an isomorphism.
So what is happening is that by definition the action on an arbitrary element of the vector space is defined to be the linear extension: $\sigma\cdot (\sum\lambda_ie_i):=\sum\lambda_i(\sigma\cdot e_i)$.
You can check all the axioms to confirm that indeed, the originally defined group action on the basis has now been promoted to a group action of $S_3$ on the entire vector space spanned by the basis.
This "lifting" of the group action is what you were missing, I think.

There's another way to see how this happens. You can rephrase a group action of $G$ on a set $X$ to be a homomorphism from $G\to Sym(X)$.  
If $X$ is the finite basis of an $n$ dimensional $F$ vector space, we have another group homomorphism from $Sym(X)\to GL(n,F)$. Composing these two homomorphisms we get a group homomorphism from $G\to GL(n,F)$, and that defines an action of $G$ on $span(X)$.
It says, essentially, that every element of $G$ acts like a linear automorphism on $span(X)$ (very nice elements of $Sym(span(X))$!)
A: A vector space of dimension $3$ with the underlying field $F_3$ is isomorphic to $F_3^3$. A standard basis for this vector space is $\{e_1=(1,0,0)^{T},e_2=(0,1,0)^{T},e_3=(0,0,1)^{T}\}$. As we know, A linear map $f$ from the vector space to itself can be represented by a matrix of this form \begin{bmatrix}
    .       & . & . \\
    f(e_1)       &f(e_2) & f(e_3) \\
     \
    .       & .&.
\end{bmatrix}
Since a permutation just permute the basis vectors, The resulting linear maps are exactly the 3by3 permutation matrices. Foe example, the permutation $(1,2) \in S_3$ can be represented by the matrix 
\begin{bmatrix}
    0      & 1 & 0 \\
    1       &0 & 0 \\
     \
    0      & 0&1
\end{bmatrix}. So you can really think of your group as the set of permutation matrices and a group element acting on the vectors by left multiply the matrix with the vector. For example, $(1,2)$ act on the vector $(1,2,2)^{T}$ gives the vector $(2,1,2)^{T}$.
A: The elements of $V$ simply consist of triples of elements from $F_3$. So a typical element of $V$ would be $(1,0,2)$ or $(2,1,2)$ etc. Addition of vectors in $V$ follows the rules of addition in $F_3$ so $(1,0,2)+(2,1,2) = (0,1,1)$. Scalar multiplication of vectors follows the rules of multiplication in $F_3$, so $2(1,0,2) = (1,0,2) + (1,0,2) = (2,0,1)$.
I think there is an unstated assumption that each action $\sigma$ will be linear on $V$, so 
$ \sigma(e_1 + e_2) = \sigma(e_1) + \sigma(e_2) = e_{\sigma(1)} + e_{\sigma(2)} $
