Is there a way to make smooth interpolated motion with the graph editor? I do animating entirely manually, sometimes 1 frame by 1 frame. Then, I discovered the graph editor and thought it might possibly have some use, though I have yet to see how.

Suppose I make a basic animation of a cube moving in a diamond-like shape. Is there an easy way to use the graph editor to turn that diamond motion into circular motion so that the cube now looks like it is orbiting something? Because otherwise, even if you add extra frames manually, it will still look jagged, just a little bit less so.
Yes, you can do this with the graph editor:



If your goal is to exactly follow a particular path or route, then there are certainly better ways to go about it (such as using a follow path constraint, as mentioned by moonboots), but you can in fact make a perfect circle using the graph editor without any keyframes at all; using sine modifiers:



Here I've set a sine modifier on both the X and Y channels with Phase Offset on the X channel of $\frac{\pi}{2}$ (or 90°). The Amplitude and Phase Multiplier are only set in order to match the example from above.

The follow path constraint is easier to control when you are looking to.. well, follow a particular path (big surprise), but often you don't care too much about the exact path but more that each motion flows smoothly into the next. The graph editor gives you fine control over exactly how that happens.