Cut drywall with a utility knife using a drywall square as a guide. Score the sheet then snap it back to break it. The cut the paper on the back side. You can also use a drywall saw to cut the drywall around obstacles. You can use either nails or screws to fasten drywall.
Screws are stronger.
Use one screw every 12 inches along the framing. There are special drywall screw guns which let you sink the screws a little below the surface. Nails are probably easier to use. Use two nails every 12 inches along the framing. Drywall hammers have a special head that sinks the nails below the surface.