Let $X$ be a Banach space. Then the weak topology on $X$ is defined as the coarsest topology such that the linear functionals in $X^*$ are continuous. A sequence $x_n \in X$ weakly converges to $x$ if and only if $$f(x_n) \rightarrow f(x), \quad \forall f \in X^*. \tag{1}$$
Similarly, the weak* topology on $X^*$ is defined as the coarsest topology such that the evaluation functionals $x(f) \mapsto f(x)$ are continuous. A sequence $f_n \in x^*$ is said to converge to $f$ in this topology if $$f_n(x) \rightarrow f, \quad \forall x \in X. \tag{2}$$
(1) and (2) are often given as definitions, but I would like to get a better insight into how these definitions follow from their respective topologies. Does this follow from their neighborhood bases at 0? For example in the weak topology such a neighorhood takes the form $$N(f_1,...,f_n;\epsilon)=\{x :|f_i(x)|<\epsilon, \text{for }i=1,\dots,n\}.$$ If so, how does this neighborhood base imply (1), and similarly for the weak* topology?
Well, the weak topology is in particular a topology where all functionals are continuous. So if $x_n\to x$ in $X$ under the weak topology, it is necessarily the case - by continuity of $f$ - that $f(x_n)\to f(x)$ in $\Bbb R$ (or $\Bbb C$), for any $f\in X^\ast$.
If you like, you can see that explicitly via the explicit basis. Fix $\epsilon>0$ and $f\in X^\ast$. $N(f;\epsilon)(x)$ is a weak neighbourhood of $x$ and thus contains $x_n$ for all large $n\ge N$. That means exactly that $|f(x)-f(x_n)|<\epsilon$ for all $n\ge N$... by arbitrariness, $\lim_{n\to\infty}f(x_n)=f(x)$ follows.
The same remarks apply for the weak-$\star$ topology.
An interesting question is whether or not property $(1)$ fully characterises the weak topology. I don't know the answer.