One Small Favour/Quick guide

Karamja

 * Talk to Yanni Salika in Shilo Village.
 * Talk to the jungle forester south of the village. You will receive their blunt hatchet. (4, 1)
 * Go due west and up a ladder to talk to Captain Shanks on board Lady of the Waves. (1)

Asgarnia and Misthalin

 * Speak to Brian in Port Sarim. Make sure he takes the blunt hatchet before you leave. (3, 2, 4)
 * Speak to Aggie, the witch in Draynor Village. (1, 3, 2)
 * Go to the H.A.M. Hideout west of the Lumbridge cow field, pick the lock on the Lumbridge mine, and climb down. Head to the southern cavern and speak to Johanhus Ulsbrecht. (4, 3, 2)
 * Talk to Fred the Farmer near the sheep's pen in Lumbridge . (2)
 * Talk with Seth Groats near the chickens' pen in Lumbridge. (2)
 * Take 3 steel bars to Horvik in the platebody shop in Varrock, who is ill and needs a herbal tincture. (2, 3, 2)
 * Speak to the Apothecary. (1, 5, (wait), 2)
 * Go to Gunnarsgrunn and talk to Tassie Slipcast. (1)
 * Enter the Dwarven Mine and find Hammerspike Stoutbeard to the west. Ask him about Tassie Slipcast. (4, 2)
 * Go to Taverley and talk with Sanfew. (3, 2, 3, 1)
 * Make a Guthix rest (vial of water, harralander, and marrentill), travel to the top of White Wolf Mountain, and speak to Captain Bleemadge.
 * Give him the potion. (2, 4)

Kandarin

 * Travel to Catherby and speak to Arhein. (5, 1)
 * Go to Seers' Village and speak to Phantuwti Fanstuwi Farsight. (4, 5, 1)
 * Enter the Goblin Cave and head north until you find Petra trapped in rock. Right-click and select "Search".
 * Go to Ardougne and speak to Wizard Cromperty. (4, 2)
 * Whilst you are in Ardougne, pick up five pigeon cages south of the north-western bank.
 * Go to Port Khazard and talk to Tindel Marchant on the southern pier. (2, 1)
 * Go to the Feldip Hills and talk to Rantz. (1, 1, 4)
 * Go west until you see Gnormadium Avlafrim and a gnome glider. Talk to him (4, 1) and search each landing light for a gem, cut it, and put it back in the landing lights (from west to east is sapphire, opal, red topaz, jade). If you crush one talk to him again to buy a replacement (4). Talk to the gnome again. (4)

Completing the favours

 * Go back to Rantz to receive a comfy mattress. (1)
 * Take the mattress to Tindal in Port Khazard, who gives you iron oxide. (2)
 * Take the iron oxide to Wizard Cromperty in Ardougne, who gives you an animate rock scroll. (4)
 * Cast the spell on Petra in the Goblin Cave and kill Slagilith when he appears.
 * Cast the spell on Petra again.
 * Go to Seers' Village and speak to Phantuwti. Go through the options until you threaten him, then access the roof via the ladder in the building to the east.
 * Use a hammer with the weather vane and search to get three broken vane parts. Go to an anvil with a bronze bar, an iron bar and a steel bar and use the parts of the weather vane with the anvil. Go back to the roof of the house and put the weathervane pillar, the directionals and the ornament back.
 * Talk to Phantuwti again and he will give you the weather report.
 * Take the report to Arhein in Catherby.
 * Talk to Captain Bleemadge on top of White Wolf Mountain.
 * Talk to Sanfew in Taverley.
 * Talk to Hammerspike Stoutbeard in the Dwarven Mine. Kill his minions. Talk to him again.
 * Talk to Tassie Slipcast in Gunnarsgrunn (Barbarian Village). Form the pot lid at Potter's Wheel. Fire the pot lid in the kiln, and then use it on a pot to get an airtight pot.
 * Go to the apothecary in Varrock and give him the pot.
 * Go to Horvik and give him the herbal tincture, breathing salts and five pigeon cages to get some chicken cages.
 * Go to Seth Groats in the farm in Lumbridge and give him the chicken cages.
 * Go to the HAM headquarters and talk to Johanhus.
 * Go to Draynor Village and talk with Aggie the witch.
 * Go to Port Sarim and talk with Brian, who gives you a sharpened hatchet.
 * Go back to Shilo Village and speak with the jungle forester, who gives you red mahogany logs in exchange for the hatchet.
 * Speak with Yanni in Shilo Village.
 * Quest complete!