Thanks, I've been doubting whether to buy a oscilloscope or not for about two years and your comment pushed me over the edge. I just ordered a Siglent SDS1104X-E. I have the same lack of intuition for electronics and I hope seeing the behavior will help me build a better mental model.
It goes into detail about how the ground lead on the probes is ground and the fact that you have to pay attention to where you are connecting them otherwise you can create a short circuit. Note that all the probe grounds are connected together, so connecting two ground leads to two different circuit nodes effectively shorts those two nodes together.
