Let me describe the standard way to do this kind of computation. The key claim is that to obtain this kind of formulas, one can pretend that the vector bundle is a sum of line bundles (even if it is not true in general). This claim is called the splitting principle and can be justified but I will not do it here (unless if explicitely asked). Let me just show how it can be used in practice.

We have our rank two vector bundle $E$. Let us pretend that $E=L_1 \oplus L_2$ where $L_1$ and $L_2$ are line bundles. Then $c_1(E)=c_1(L_1)+c_1(L_2)$ and $c_2(E)=c_1(L_1)c_1(L_2)$ are the elementary symmetric functions in $c_1(L_1)$ and $c_1(L_2)$.

We have $Sym^n(E)=L_1^n \oplus (L_1^{n-1} \otimes L_2) \oplus \dots \oplus L_2^n$ hence

$c(Sym^n(E))=\prod_{i=0}^n c(L_1^{n-i} \otimes L_2^i)=\prod_{i=0}^n (1+(n-i)c_1(L_1)+ic_1(L_2))$

This expression is a symmetric polynomial in the variables $c_1(L_1)$ and $c_1(L_2)$ and so can be written as a polynomial in the elementary symmetric polynomials in these variables, i.e. as a polynomial in $c_1(E)$ and $c_2(E)$.