What would be the best gravel to use for installing a water drainage?

You could spend a great deal of money on gravel, which might still shift. Can you plant any perennial ground covers in some of this space, but far enough away from the oak? If people use the sidewalk and lose their balance/fall in the gravel, can you be legally liable?

The smaller the gravel/rock, the quicker it will plug up with debris, leaves, etc. Your best bet would be river rock, which will allow the water to drain and still not get restricted with litter.

I have used a blend of river rock and lava rock. River rock is smooth and can shift while lava rock is coarse and will hold.