People new to hanging drywall sometimes measure a doorway opening, cut the drywall to fit around the edges of the door frame and then hang the board. In other words, to surround the doorway opening, they cut a rectangular hole in a board of drywall. A much easier method is to place a full board over the doorway, drive in screws around the borders and then go outside the room. Use a utility knife to score around the inside border of the door frame from the opposite side. Break along the score lines to snap off the excess.
If you find cutting with a utility knife is too slow or imprecise, a drywall saw might be preferable. Its jagged teeth cut through drywall quickly and evenly. But the most effective tool is a drywall router, which has a pointed cutting bit. You don't need to go outside the room to make the cut with such a tool. Just squeeze the trigger or depress the switch to turn it on, and poke the router bit into the drywall near the edge of the door frame. Steer it close to the edge of the door frame. Use the inside of the door frame as a guide for the bit, allowing the router to cut the opening.
Sometimes the drywall edges around a doorway are rough and uneven. Typically, this shouldn’t be a concern because the wood trim or door jamb will cover the uneven edges. But if necessary you can smooth the edges with a drywall rasp, which is similar to a cheese grater. A drywall rasp is also convenient for reducing the size of a board by a tiny amount.
After the drywall is installed, most people install wood door trim over it. If that is your plan, drive in the drywall screws no farther than 1 inch from the edge of the door. This way, the wood trim will cover the screws, meaning you can skip coating the screw heads with joint compound to hide them.