The most common methods I've seen to find a line of best fit are Least Squares regression and median-median. Are there other good ways? Is there a way to minimize the absolute value difference and find a line of best fit that way? Or to find the distance straight to a line instead of the vertical distance to the line? Thoughts?
Minimizing the sum of absolute differences is quite common, as Nick Cox suggests, it's often called L1 regression or Least absolute deviations regression; it's also a specific case of quantile regression and many posts here relate to it.
The orthogonal distance (what I assume you mean by "straight-line distance") would correspond to a particular case of Deming regressing, itself a particular case of the total least squares line, called orthogonal regression, which will give the line of the first principal component.
Some discussion of robust regression (including some comparison of Theil-Sen and L1 regression) is here.
There's some interesting discussion relating correlation measures to straight-line fits here