Although there are private companies that take your trash to the landfill, most of the Weyers Cave residents just take it there themselves.
The closest dump to most Weyers Cave residents is in Mt Sidney, at the intersection of Route 11 and Route 696. It is open from 7 am to 5:30 pm in the winter, and during Daylight Savings Time is open until 6:30 pm. For more information, here is the link to the Augusta County website for Recycling Locations.
Used but fixable computers and other electronics can be dropped off at Equipping the Saints or GoodWill locations.