What tool do I use to cut out a hole in our drywall and ceiling (pic)?

There's a special saw, a handsaw, for this. Any building supply place will have one. You give it a good whack with a hammer to punch through the paper, then start sawing. Wear a mask for working overhead, the drywall dust will fall all over your face. A jigsaw works too but if you don't want to invest in one, the drywall saw is cheaper.