Should the exhaust to the range hood be to the outdoors

YES

If installing a range hood that has duct work connected to it (any hoods that do NOT recirculate the air back into the room), to properly exhaust air, be sure to duct air outside. 

DO NOT vent exhaust air into spaces within walls or ceilings or into attics, crawl spaces, or garages.

A build up of humidity and grease could cause damage and/or a fire risk to your home.