I am to find a squence from $c_{00}$ space (space of sequences with only finite non-zero entries) that is weakly convergent but not bounded.
My idea is to take $c_n=(0,..., 0, n, 0, 0,..., 0)$. My guess it would be weakly convergent to $0$ - but I am not sure.
No, a weakly convergent sequence on a normed (even non-Banach such as $c_{00}$) space is always bounded.
This follows from the uniform boundedness theorem: suppose $x_n \rightharpoonup x_0$ in a normed space $X$. Consider the operator \begin{align*}J:X&\to X^{**}\\ x &\mapsto Jx:f\mapsto f(x),\;f\in X^*\end{align*} It is not difficult (although you need the Hahn-Banach theorem) to verify that $\|Jx\|=\|x\|$ for all $x$, so $\left\{x_n\right\}$ is bounded iff $\left\{Jx_n\right\}$ is. But for all $f\in X^*$ we have $$|Jx_n(f)|=|f(x_n)|\to |f(x_0)| $$ Thus $\left\{Jx_n\right\}$ is a pointwise bounded family of bounded linear functionals on $X^{*}$, which is always a Banach space. By the uniform boundedness theorem, $\left\{Jx_n\right\}$ is bounded in $X^{**}$, and therefore $\left\{x_n\right\}$ is bounded in $X$.