Archery is intoduced to the player when he talks to Robin. Robin is located to the far left of the Evergrind Fields: South inside the house. When the player talks to him for the first time he will give him/her a bow and 20 arrows. He will also tell the player about his minigame. The minigame requires the player to hit fake slimes (green, red and blue) that pass by at various speeds depending on the colour. The player does not have to play the minigame and it costs 50g for every try.
When the player starts the minigame he will be given 15 arrows to shoot at the slimes that pass by on the shooting range. To beat the minigame the player has to score above 10000 points, this is done by hitting only the blue slimes at the back without missing and by keeping up the score multiplying (I think you have to hit something within 5-6 seconds to keep up the multiplying). There will appear 2 bees that gives the player a big amount of points (first one appears after reaching 8000 I think and second one appears around the 10000 mark). The highest score reported so far is 13125 and requires the player to hit 13 red slimes and 2 bees without losing the score multiplying.
Hitting a green slime increases your multiplier with 25 points for each green slime hit. Green slime gives 100p without multiplier.
Hitting a red slime increases your multiplier with 50 points for each red slime hit. Red slime gives 200p without multiplier.
Hitting a blue slime increases your multiplier with 75 points for each blue slime hit. Blue slime gives 300p without multiplier.
So hitting 2 green slimes and 1 red slime will result in an extra (25+25+50) 100 points on top of the base points.
E.g. You hit 5 blue slimes in a row. The 6th blue slime will earn you (75*5+300) 675 points. Hitting 6 blue slimes in a row earns a total of 300+375+450+525+600+675= 2925 points.
Archery in combat can be used in different ways, it can be used to push enemies like Rabbys or Brawler Bots back or to snipe the ranged monsters like the Wisps or the Lantern Jacks. It can also be used to lower bosses hp as the bow can be spammed very quickly. The bow can hit more than 1 enemy if the enemies are stacked on top of each other. This can be very useful in the arena or in Arcade mode as you don't want to take any damage.