Weak convergence of continuous functions
For sequences, there is the following simple characterization:
Proposition. Suppose $f_n, f \in C_0(X)$. The following are equivalent:
$f_n \to f$ weakly;
$f_n \to f$ pointwise and $\sup_n \|f_n\|_\infty < \infty$.
Proof. $1 \implies 2$: The evaluation maps $f \mapsto f(x)$ are continuous linear functionals, so $f_n \to f$ pointwise, and $\sup_n \|f_n\| < \infty$ follows from the uniform boundedness principle.
$2 \implies 1$: If $f_n \to f$ pointwise and $\sup_n \|f_n\| < \infty$, then for any signed or complex measure $\mu$, we have $\int f_n \,d\mu \to \int f\,d\mu$ by dominated convergence, using $\sup_n \|f_n\|$ as the dominating function. (For positive measures $\mu$, this is the classical dominated convergence theorem. For signed measures, use the Jordan decomposition $\mu = \mu^+ - \mu^-$. For complex measures, take real and imaginary parts.) $\quad\square$
This characterization does not work for nets. For example, a weakly convergent net need not be bounded.
You can find this result (for $X$ compact, but the proof is the same) as Example 2 of IV.5 of Reed and Simon, volume 1 (Methods of Mathematical Physics: Functional Analysis, Revised and Enlarged Edition).