Forage and Woodcutting are both under the Agriculture Skill. You can either capture a Student NPC and donate him to the Goblin households where he will offer to teach it. Or you can visit the Brothel on Saturdays and head to the second floor where there will be a pink haired demon girl NPC that sells all the Skill Manuals.
Battle EXP can be spent to level up the Athletics, Black Magic or White Magic skills. To level them up just open the Skill Manual in your inventory and it will bring up the interface for them. Though, if you don't have those Skills then you can't really use it. You can also merge Craft EXP with Battle EXP to get Combo EXP which is used to level up the Perception Skill if you have it.
If you don't have the Athletics Skill, you can get it as a reward from a Goblin Guard NPC in the Goblin Tower on the 2nd Floor after you bring her 10 Honey from Hornets.
Hope this helps!