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I know that stress is concentrated in the fillets, but what about the average stress along the entire hole? We can choose a circle, an ellipse, or a square with fillets.
For which shape would the average stress be minimized? Also which has the highest/lowest peak stress?
It depends on the loading conditions.
It's important to note that the stress concentrations can be intuitively understood as created by the load's need to deviate from the hole. The larger the width of the hole, the more the load needs to move to go around it.
If it's a uniform bi-axial load (your part is under tension/compression in both the X and Y directions), then the best shape is a circle, since it treats all directions equally.
However, if it's a uni-axial load, then an oval with the major axis in the same direction as the load (i.e. a vertical oval in @NMech's image) would be better. That's because the load will need to move less to the left or right to get past the hole. But the oval isn't the best shape: that'd probably be a very thin slit (think a cat's eye).
The average stress across the cross-section (not around the whole) will be $$\sigma = \frac{F}{A}$$
To my understanding, the stress concentrations redistribute the path of the forces, thus increasing locally the stresses.
Figure 1: flow path (source Fracturemechanics)
The highest observable stress is important because, from that point failure will initiate, and in most cases it will propagate.
