How do we know the true value of a parameter, in order to check estimator properties? For example, we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate. However, if we already know the true value of the parameter, why are we trying to estimate it? I know there's something I'm missing here, like some theoretical true value that we can find, otherwise I don't see how we can check for unbiasedness, consistency etc.

Phil has given a very good answer regarding the unbiasedness of the estimator. But I think your question may actually stem from another confusion:
You seem to mixing up the estimate and the estimator. The estimator is a general rule how to calculate a specific value from a given sample. This sample-specific value is called the estimate.
If you use the estimator $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$ to estimate the unknown population mean $\mu$, then you have an unbiased estimator because $E(\bar X) = \mu$. Applying this estimator to a given sample, you get a specific value $\hat X$ (the sample mean), which is called the estimate. However, this does not mean that $\hat X = \mu$, i.e. the actually estimated mean $\hat X$ for a given sample is not necessarily equal (or even close) to the true value. To get an idea how well we estimated the true value, therefore we usually express our uncertainty of the estimate, e.g. via confidence intervals.
