Another Piece of Something
The Eel can be found swimming in the southern pond of the Field map. The Iwana can be found near the Waterfall in the Stream, and the Smelt is in the pond on Benkei's island in the Mountain. Bring all 3 fish back to Serika and she will give you her Piece of Something.
Naoko Fan Club number 1 fan
The Monster Bass can be found in the darker water of the ponds in the eastern corners. A Spinner lure works well. Catch the fish and bring it to Naoko. She is to the south of the Cooking Contest, just past the bridge. Give the fish to Naoko and she will give you her autograph. Take the signature back to Number1 and he will reward you with a Giant Basket. Now you can hold 50 fish! Don't forget to select it from your Equipment menu.
Help the science class
The Killifish are in the waterways between the rice paddys. You can catch them with a bait rod and Insects. Bring all 6 of them back to the class and Miss Rumiko will give you the Loach Stew Recipe.
The Straw Millionaire, part 3
If you decide to give her the Vegetable Set, she will give you a Bracelet in exchange.
Run Away Brother
Kazuki wasn't kidding when he say his brother was far away. You'll find his brother Mizuki all the way in the northeast corner of the Swamp Map! Mizuki ran away to get back at their father, but he lost his way and couldn't make it back to the Field. Mizuki is afraid of what their dad might do when he gets back, so you two decide to make a piece offering of Broiled Prawn.
You can run back to the Field to get one, or go to the southwest corner of the Swamp and fish among the floating logs in the waterway. Bring the Prawn back to Mizuki (you don't have to cook the dish yourself) and Mizuki is now ready to go home.
Back at the Field map, Kazuki will reward you with the Amago Saute recipe.
Raccoon Dog moves in
The little guy will then continue to ask questions over and over. If you answer all 10 of them correctly the Raccoon Dog will tell you that it's moving to live in the Inn. If you want to visit the animal you can go into the Inn and talk to it near the staircase. Tanuki will say a word or two and then go back to doing nothing. You won't receive any reward for making Tanuki move to the Inn, but he is awfully cute sitting there. You'll see Tanuki in all of the Inns you visit.