As you already know, if you increase bat speed, you increase the power or distance that you can hit the ball.
But, how do you increase bat speed?
There are two thoughts on the swing. Linear Mechanics vs Rotational Mechanics.
Linear is a forward movement toward the ball. While rotational is a spinning movement.
Happy Gilmore would be an exaggeration of linear mechanics. George Brett would also be an example.
This article makes the argument that rotational mechanics has the biggest upside to bat speed.
Then it goes on to link over to a Ted Williams advocate on a rotational swing.