This guide was written by: MrsKittyHawk, thank you!
Since most people do not know the exact location of the festival, nor any other useful information, I am presenting everything that is known here. It is not an extremely detailed guide, so you might have to do some of the work yourself, but don't worry, it's not that hard.
Full Moon Festival ~ Location
The festival takes place in Artecia in the bottom left corner, just take a look at the arrow on the map in the picture.
Beginning the Festival
In order to start the festival, you want to check in with Full Moon Master Chef Sherf.
He will give you a quest that you need to complete. In addition, you should talk to the Full Moon Cookie Aunt Lunaria, she is responsible for the cookies you will need to eat for the "Addicted to Full Moon Cookies" achievement and the corresponding "Full Moon Cookie Fanatic" title. You need to eat 10 of these cookies, but the crunch of it is, that they have a 24h cooldown.
The Recipes and the Chefs
There are 8 recipes and 8 chefs. Each of the chefs will give you a recipe that you have to prepare. You use them just like normal recipes from your inventory. For some of these recipes, you might need Doran Grass, plates of Minced Meat and other special ingredients that you can get at the Full Moon Festival Ingredient Merchant just across from the Full Moon Cookie Aunt. For these ingredients, you need a Full Moon Festival Ingredient Coupon. Keep in mind that different ingredients have different prices.
1 Coupon: High Quality Steak, Softspine Fish, or Flyfang Thigh
2 Coupons: Fresh Banti Tongue or Sourbite Berryfruit
6 Coupons: Spikebrew or Pristine Spring Water
16 Coupons: Sweet Almond Paste or Icing Sugar
You will earn these coupons by preparing the recipes and bringing them to the right customers at the tables. You will receive a refund for any used ingredient (once for each Doran/Mince Meat recipe. After the first cookie has been eaten, each further cookie will offer 16 vouchers.
The chefs each offer a recipe that you will need, but some of them are not there all day. Here is a list of the chefs, the recipes they offer, and when they appear (time of day):
Shavio the Pottage Wizard -- Day -- Oziak's Source
Fisher the Fish Master -- Day and Evening -- Tarrock's Gift
Miller the Meat Cook -- Morning and Evening -- Blaze of Samoor
Marcello the Head Chef -- Evening and Nighttime -- Lysaia's Wisdom
Sabina the Salad Master -- Morning and Evening -- Balian's Forest
Domitira the Pastry Chef -- Nighttime -- Karuna's Delight
Billberg the Drink Maestro -- Nighttime and Morning -- Water of Styx
Vianna the Appetizer Expert -- Day -- Logath's Balance
Once you are done preparing a meal, you will receive the achievement "Full Moon Festival Participant". If you prepare all the meals, you will receive the achievement "Full Moon Festival Professional Chef" and have earned the title "Full Moon Festival Top Chef". If the meals were consumed, you will receive the achievement "Gourmet" and the title "Full Moon Festival Big Eater". Once you have done all that and eaten the 10 cookies as well, you will have completed all the Full Moon Festival activities and receive the corresponding achievement and the title "Full Moon Festival Maniac".
You can find information about the chefs on the bulletin board next to the ingredients merchant as well.
Happy cooking for all of you.
These are possible "transformations":