When you are on the World map, head straight ahead and enter the area marked with a '?', this is Madain Sari. Eiko lives here along with a bunch of moogles. After Eiko is done asking Zidane a barrage of questions, search behind the cracked portion of the fountain to find the Libra Stellazzio. Move next to the bench that Dagger is sitting on and open the chest to receive a Tent. Exit this area, talk to the moogle and Quina will appear.
Head back to the area with the fountain and an ATE will start. Give the moogles their tasks, I am not sure if it matters who you choose for the different tasks so just select anyone. Once you are back in control of Zidane head to the left to find Vivi. When they are done talking attempt to enter Eiko's house, Morrison will stop you from entering and he will take you to see the Eidolon wall, which you were denied access from by the moogle standing guard.
Follow Morrison through Madain Sari, another ATE should activate. When you are asked how many to account for, choose 11 people. When asked whether or not to add an oglop to the food, choose not to. Choose to assist the moogle who is fishing. After the ATE is over continue following Morrison to the Eidolon Wall. Moco reluctantly retires from his position so Zidane and Morrison can head inside. Zidane chooses to go back and get Dagger so she can see the wall as well, pass through the bottom right path and you will find Dagger there. Talk to her and she will join your party, then head back to the Eidolon wall.
When you are finished looking at the wall, leave Dagger behind and exit the area. Another ATE should activate, select the option to ask Quina for advice. Quina heads off to ask Vivi for help with the cooking. Once you have gained control over Zidane again, head towards Eiko's house and enter. After they are done eating, open the chest to receive an Ore, move around the table and open the one behind it to find a Phoenix Pinion. Head over to the table, grab the pot and head through the doorway that Eiko passed through.
Once you are outside talk to Eiko, talk to her a few times until she doesn't have anything else to say. Search the left corner of this area and when the '!' appears search to find Kirman Coffee. Head back into the previous room and attempt to leave, sleep when the moogle asks you. You can go back to Madain Sari and talk to Morrison, you can purchase some equipment and healing items from him. Take a look at what he has and when you are finished head back out to the world map.