How do I derive the distribution of a real-valued statistic from real-valued random variables with known univariate distributions?

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Question

I want to improve my skills at finding the distribution of functions of real-valued functions of real-valued random variables. I’m familiar with the basics of the algebra of random variables as far as manipulating expectations, and also understanding the expectation as an integral. I also think I understand that a sum of two random variables is a convolution, and a product of two random variables has a distribution that can be derived from their cumulative distribution function. But, let me formalize what I am interested in.

Let there exist a collection of real-valued random variables ${X_j \sim f_j}_{j=1}^{n}$ where the univariate distributions $f_i$ are known, and let there be a population parameter $\theta$ that is estimated by $\hat{\theta}$ which is a function $g$ of the random variables:

\[\hat{\theta} = g\left( X_1, \cdots, X_n \right)\]

Let’s also assume that $g$ and $f_i$ are sufficiently smooth. But let us not assume that any subset of ${X_j \sim f_j}_{j=1}^{n}$ is necessarily independent.

While this description is not sufficient to provide an exact algorithm, I would like to know if there are some common approaches or procedures to deriving the distribution of $\hat{\theta}$.