What are the quantities in GR that one can actually measure in an experiment? It seems scalars, being coordinate independent, should be measurable. What about other quantities? For example, using only experimental observations (using some coordinates) can one find the Riemann tensor or the Christoffel symbols components? And if possible how would one go about measuring it?
You can make any measurement you want. Simply choose your coordinate system and do the experiment.
There are some practical issues. For example the components of the Riemann tensor are (in principle) measured by parallel transporting a vector round a loop, but moving something round a closed timelike loop presents some practical problems. For some quantities you'd need to do an indirect measurement e.g. measure the metric or the Christoffel symbols and use them to calculate the Riemann tensor.
I suspect you're getting mixed up between what is measurable and what is an invariant quantity. Invariant scalars will have the same value in all coordinate systems. Objects like the metric will have a representation, i.e. the individual entries in the matrix, that depends on the coordinates you choose. But it's still the same object you're measuring - it's just the way you write it down that varies with your coordinate system.