The first section of old forest is shown below on the map with chests indicated. You need to head toward the South East corner of the map. When you reach the bottom corner, Egbert will learn the Chop ability which allows you to chop some types of bushes. This allows you to get any chests you couldn't get before.
Enemies in this area are bazooka ants, tough ants, machete ants, wolves, faffs and ghosts.