The first person you will see upon entering Daminu Forest is Ampeis. Accept his quest and all other quests in this area.
You should have the following quests:
With all of these quests in hand, head out into the wilderness. Out here you will want to kill Everglade Sylphen for 5 of Sylphen Wings. While you're killing the Sylphen you will also want to kill the Sparkie's for Sparkie Luciferin.
After you complete both quests you will want to head more east over to the Shortbristle Bellepigs. Here you will need to kill 8 Shortbristle Bellepigs.
After you finish killing the Bellepigs you will want to head south of the path and find Sleeping trees for "Request of the Elim". In order to wake them up, simply right click them.
Once you're done all 3 quests return to the little camp in Daminu Forest and turn in your quests. Accept the follow up and use your "Return" ability to return to Akarios Village. Here you will want to turn in your finished quest and accept the followup. Also accept "Pernos's Robe" from Mune, who is the Armor vendor.
After you've done this exit town and head back to the farmer alongside the road. Turn in your "Light up the Night" quest to him. After you do that head back to the Fishermen by the lake. Refresh your quest at Feira and then return to Akarios Village again. Turn in your quest to Asteros which will probably ding you level 6.
Buy and new skills and accept the new quests from Polinia and Baevrunerk. Also while you're at Baevrunerk you will want to select the "Expand Cube Option" which will increase the space in your inventory. After you've done these things leave town and head back to our little camp in Daminu Forest. Back at the camp in Daminu Forest turn in your quest to Damus and accept the new one that he has.
Right near this small camp you will want to kill Grove Abex's for 5 Abex Skins. Also chances are while killing the Abex's you will find a journal which will start a quest that can be turned into Namus for some extra xp. The followup isn't worth doing, so you're better off ignoring it. Continue down the path to reach the Deforested area.