Since all of the discussion above focused on mechanics for territory counting (including the AGA video for how to use territory counting to get the area score), let me add something to discuss area counting mechanics.
With area scoring, you only need to manually count one player’s score, since the scores should add up to 361 (if playing on a 19x19 board) minus any unfilled dame. Technically (under Chinese rules, specifically), the unfilled dame points are also split as a half-point to each player, but that can be ignored without changing the result.
Note: some dame might have to be left unfilled for the sake of preserving seki. It’s also possible to leave an even number of inconsequential unfilled without affecting the result.
Before counting the score, the players should both agree which stones are dead and remove them from the board. If there is any disagreement, play should continue, and the burden of proof is put on the player claiming death, who must capture those stones to settle the dispute, and if they fail to do so, those stones would be considered alive.
To count one player’s score, you need to count their total area occupied (living stones + surrounded territory). First, count the territory, and you can use the same stone rearranging tricks mentioned above to form territory into rectangular regions to make for easier counting. However, you can also add and remove stones (of that player’s color) to make the territory counting more convenient. This does not affect the score, since when you reduce territory by one point, you are also adding a point by having another stone on the board, and vice versa. After counting the territory, count that player’s stones on the board (including the adjustments made during territory counting). These stones can be formed into groups of 10 or rearranged into rectangles to ease counting.
On small boards like 9x9, control over the board might wind up split in half, and one may only need to count along a diagonal or a center line to figure out the score difference.