My question is when you probe an arbitrary circuit, and you see
something that you weren't expecting, how can you determine if its
coming from the scope/probing or from the circuit itself?
Its not always so easy, as there are a lot of ways to get the same suspicions display. The best thing to do is cut the problem in half, starting by seeing if the same scope, with the same wires, probes, and settings, will do the same thing connected to another source. For example, if there are indeed pulse edges that are damped, have overshoot or ringing, see if its still true connecting to a known signal source of similar frequency. To start, many scopes have square wave calibration outputs. As long as your probe is on its X10 setting, you should be able to trim out overshoot or damping with the adjustment within the probe (usual via a small screw). If the frequency being measured is much higher than the calibration source, and you don't have an alternate, you may be able to deduce whether its a scope / probe problem by seeing if multiple points in the DUT show similar ablations.
Another comparison, if the DUT's power source is ground isolated, you can use two probes in differential mode, one connected to the DUTs common 0 Volt reference, the other to your test point. Often this will produce wildly different waveform qualities, depending on how much of the original problem is caused by grounding. Isolate your scope from AC power ground too. But if nothing changes and the suspicious display continues, you're at least a step closer to eliminating the scope.
But only to a point! Probe quality and the impedance of both the scope and probes can be a big factor, and at some point the limits of your equipment can bring you too close to the point where the famous uncertainty principal (which boils down to the impossibility of taking a measurement without affecting the measurement). I remember trying to troubleshoot DRAM< circuits, in which sensitive RAS/CAS lines were causing problems from ringing, and I was trying to choose the optimum damping resistor to control it. In the end I had to borrow a better scope, because the one I was using was adding to the problem. You too may need to procure at least some alternate probes, again to see if there is any change.
Now if the problem is more like superimposed noise (power supply switching or ripple noise, or other switching noise), then of course it will persist through all the above recommended checks. But a noise issue should be a little easier to trace, because it if is coming from the DUT, you will likely be able to find stronger and weaker sources of noise with the same visual signature, as you probe around the circuit. If the noise is present everywhere at the SAME levels, then you're probably back to the scope.
I'm sure my answer is probably little more than an "I feel your pain" response. But hopefully it will help you understand what you're up against, and something here will trigger a path to a solution.