In the Euphonia's Challenge quest during the tutorial, jump to the very top and onto the wooden ledge that borders the room. In the corner is a chest that contains Starter Repair Kits and Start Healing Kits.
In the quest where you go to Sigmund's cellar to find the source of the unholy artifact, there are two chest to find. The first is in the room where there is a big hole in the ground. Go up the ladder, open the chest, and get Aida's necklace. After the quest, talk to Aidas in the corner of the tavern to get 3 GP. To find the second chest, after you use the altar on the quest, check the room that has Jocoby the evil cleric. You should find a chest that contains some gold or equipment.