Very quick and short clarification:
Knowing $R_{abcd} = - R_{bacd}$, is this symmetry true for for a,b components (i.e $R_{acbd}=-R_{bcad}$) or for the first two 'slots' of the tensor (i.e $R_{acbd}=-R_{cabd}$). This is specifically for the Riemann Tensor if that matters, although I assume there is a strict convention.
I am sure this is a very basic fact about how tensors transform but I haven't found a clear "this is how it works".
The letters are just there to keep track of each entry, they generically do not represent anything by themselves (unless you are prescribing them a specific value, such as $a = r$, where $r$ is the radial coordinate). As a consequence, the symmetry applies to the slots, not to the entries.
In my opinion, Geroch's notes on Differential Geometry (available for free on his website) are particularly good to get a grip with this notation. In Chap. 4 he discusses how abstract index notation works in detail. In short, the indices play a role similar to that of the little arrow when we write $\vec{v}$, but in here they also keep track of slots.
