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Best answer as chosen by Asker / Ratings
1. Ship a truffle (super tedious, they're around the Mora trees in fall, you have to till the soil for one).

Master Angler- catch 100 fish (don't have to be different kinds).

Lost Child- Another tedious one. You need to befriend Ronald, Meryl (2-3 hearts, not sure which)and Tim (2 hearts). Meryl comes when Ronald upgrades the Paradise orchard, around 2-3 hearts. Tim comes when Tai does. Then go into your house at night and the Mayor will come to your door and tell you the kids are missing, you get a cutscene, then the note!

Art Note-get the aging pot and picture. Get the aging pot from befriending Saibara (ship 10 clay to get him). He needs to be at 3 hearts and he will ask you to run an errand for him. Do it and you get the aging pot. The picture you get from befriending Henry (plant 3 trees and go to Sunny Lake to get him). Either wait until he's at 8 hearts and he gives it to you, or get him to 5 hearts and purchase it from the Hearty Lyla for 500g. 2. Apparently Gwen needs carrots to cook or something so ship a ton of them. If you've got enough cash, you can always just buy them from the seed place. Hope this was detailed enough! :)