I have 4 wires on a new switch, 2 of them going to ground. One green and one silver (bare) coming from the bracket of the switch. In the electrical wiring of the house I have a neutral wire, what do I hook that wire to?
I need a little more information on the wiring to give an answer. Typically the white neutral wire is bonded to ground, but only at the panel. If this is a switch box, the white wire could be bringing power to the box or returning power back to a light/outlet/switchable device. Essentially, power has to feed "into" the switch with one wire and back "out" through another. It's supposed to be marked/taped though to designate this.
Alternatively, it could simply be an unused neutral leg that was brought in with the hot leg and should just be capped if not used. Newer building codes require you to run a neutral leg to switch boxes so that things can be "run" from the box. For example, you would need the neutral to install an outlet/switch combo at the switch box. Or using a motion sensing switch might require a neutral leg as well.
Can you provide a picture or some history of what type of fixture was in the box?
Neutral (white) doesn't go anywhere on a plain old fashioned switch. It is used by smart switches.
Based on the way the house is wired... If the switch is "on the way" to the light, then the neutral passes through the switch on its way to the light... that is to say, it is spliced to another white wire, probably with a wire nut. If the switch is on an endpoint past the light, then the neutral is reserved for the future (when you or the next owner decides to use a smart switch) - just cap it off for now with a wire nut.