How can I calculate standard deviation, skewness and kurtosis once I
know the PDF of a distribution?
You can either calculated the moment generating function of the PDF and then use the derivative rules to calculate all the individual moments or you can calculate the individual moments using the expectation formula.
Recall that E[g(X)] = Integral (over the real line) g(x)f(x)dx where f(x) is the PDF function.
If the distribution is not continuous, then replace an integral with a sigma (i.e. summation).