Put in a new battery, turn the key on, hit the start switch, nothing. Light doesn't go on either when I turn the key on. If I try to start it with a charger attached will I kill the battery or blow a fuse on the quad?
Any help is much appreciated.

 

Forget the charger. You will probably need a meter. They're so cheap these days that there is no excuse for not having one. Then we can follow the battery voltage through the starter motor interlock circuitry and see where the connections break down.

But first let's eliminate the easy stuff and do some rudientary tests:

1) Have you checked your fuse?
2) When you turn on the ignition and apply the brakes, does the brake light light up?
3) Set your parking brake and/or set your quad into neutral. Jump across the starter solenoid screw terminals using the shank of a scredriver. Does the starter motor turn?