I will be giving my view only on matched grip here.

If you want to learn traditional grip (marching drum style) I suggest approaching a teacher experienced in this area.

I hold the stick between my index finger and my thumb. These two fingers are the main pivot.
                                   
Notice where my fingers are situated on the stick.
The remaining fingers are held out in this photo as an example only.

My second finger grips the stick also, but it is the index finger and thumb that do most of the work.

The rest of my fingers are tucked very loosely around the stick (never tightly) The butt end of the stick is free to move in my palm and bounce or reflect off the inside of the non used fingers.

As a rule my palms face slightly down,

but become more vertical as I play faster, as in a fast rudiment or quick time on my high\hats.

I quite often play my ride cymbal with a vertical palm.