Note that you'll need Epona (see section 6) to get through this trading sequence.  Also remember you have to actually show them the item using a C-button, not just talk  to them.

1) The egg.
Talk to the chicken woman in Kakariko Village to get the mini chicken egg.

2) The chicken
Wait for a day for the egg to hatch.

3) The blue chicken
Use the mini-chicken to wake Talon in the house on the ground level of Kakariko village.  Bring it back to the chicken woman to get the blue chicken.

4) The odd mushroom
Find a sleeping man in the lost woods (just go left from the entrance) and wake him with the blue chicken.  He'll give you an odd mushroom.

5) The odd potion
Take the odd mushroom to the weird potion shop in Kakariko Village (Go to the normal potion shop and go through the hallway on the left to get to the fenced off section of the village, then go up a ladder to get to the weird potion shop.) before the time runs out. If you try using your ocarina to warp it will instantly spoil. Give the shop owner the mushroom and you'll get an odd potion.

6) Poacher's Saw
Find the small girl in the lost woods (She's where the sleeping man was, left of the entrance) and trade the odd potion for the Poacher's Saw.

7) Broken Goron's Sword
Give the Poacher's Saw to the carpenter's boss in Gerudo Valley to get the broken Goron's Sword.  You can get over the broken bridge by jumping with Epona or using a longshot.

8) Prescription
Give the broken Goron's Sword to the Big Goron at the summit of Death Mountain (NOT the one inside Goron City) and get the Prescription.  Climb to the top of Death Mountain and then look at the right edge of the plateau up there to find this Big Goron.

9) Eyeball Frog
Bring the prescription to King Zora to get the Eyeball Frog.  Remember to thaw him first by using magic blue fire from the Ice Cavern.

10) Eye drops
Bring the Eyeball Frog to the professor in the Lake Hylia lab.  If you try using your Ocarina to warp it will spoil instantly.