Note: This trip to Lindblum will involve following a strict walkthrough to get all of the ATEs, since it's very easy to miss some of them.
Once you gain control, you will be inside of the guest room of Lindblum Castle. Head to the top of the room and speak with the moogle to get a letter for Moodon, then open the chest to the left of him for an Elixir. In the bottom right corner of the room will be another chest, this one containing an Egoist's Armlet. Approach the stairs leaving the room to have Blank barge in. When presented with a choice, make sure you pick Where's Dagger?, as this will lead to an ATE later. Zidane will rush out of the room upon choosing this option, where you can view an ATE called Something Washed Ashore.
Note: This ATE will only trigger if you watched the Hallucination ATE back in Treno.
Head to the elevator next (go left and then south) and ride to the Base Level, where you'll want to go to Dragon's Gate. There will be a treasure in the usual spot here, which is a Remedy. Take a ride to the Serpent's Gate next and go to the top of the pathway for another item, this one being a Chimera Armlet. You're done in the basement, so return to the station and ride the elevator up to the upper level now. At the top, in the spot from earlier, will be Dagger (do not go in the throne room yet). After the conversation with her ends, you should receive a prompt to watch the My Bad.
Return downstairs and when you get right outside of the throne room, you should meet up with Blank again. He mentions that Cid wants to see you, so go through the doorway, up the stairs and through the door to find Cid. A cutscene will start, with two ATEs automatically playing, called Self-reproach and Deep Anger. With the meeting abruptly over, you have to find Dagger again, who is in the guest room now. Note that his is at the Mid Level, so head back there for a scene, then it will switch back to Cid.
The Three Potions
It seems the true method to reversing Cid's condition is not possible right now, but there might be another solution. This requires hunting down three potions, the Unusual Potion, Beautiful Potion, and the Strange Potion. Zidane offers to track down the potions and once outside of the throne room, you can view the My Hammer ATE. Take the elevator to the Mid Level and make your way to the air cab, as you want to ride it to the Business District next. Enter the inn across from the air cab station and go upstairs, where you want to hand over one of the letters to the moogle, Moodon, there.
Back outside, head north to the next screen and check the left side of the screen until you find the Sagittarius Stellazzio. Enter the house on the northern edge of the area to find two chests there, which will contain a Remedy and Elixir. Return outside and go into the next screen, which is the one with the shops. If you remember where the item shop was located, there are people there now fixing it up, with the proprietor standing there watching them. Speak to her (name is Alice) and ask her about the potion to have her hand over the Beautiful Potion.
Missable: An upcoming ATE requires you to have less than 50 Gil on you, so buy some consumables and preferably equipment that you can sell back. Note that equipment typically sells for around half of the price that you bought it for.
Take heed of the note above, as the ATE that unlocks is missable and it's needed for the Movie Critic trophy/achievement. If you want to work on that now, you can, but there are a few chests you'll pick up later in Lindblum that contain some Gil, so it might be better to wait until then. You have one of the potions, so it's time to track down another one, so let's leave the Business District and head on over to the Theater District. After getting off of the air cab and heading to the next screen, you should see a quick scene here with Kal and Wei, the Burmecian couple you helped out back in Burmecia, which will in turn, lead to an automatic ATE called The Rally.
With that over, find the artist's studio in the bottom right and go inside, opening the chest there for a Lapis Lazuli. Ask the NPC inside about the potion, then go to the left side to find the Strange Potion. Exit and descend the stairs for another scene with your pals from Tantalus, with Cinna handing over the Unusual Potion. Once you have control again, return to the hideout and go inside, looting the three chests there for 970 Gil, 1273 Gil, and 4826 Gil. Take heed of the warning above and return to the Business District, as you'll be purchasing a bunch of stuff now.
Before you splurge on useless junk, though, you can actually upgrade some of your equipment. The Cypress Pile is a new wand for Vivi, while the Mantra Band and Dark Hat are new pieces of equipment. There's nothing new at the Synthesis Shop, so don't bother checking. Throw away your the rest of your money, as mentioned above, making sure you are under 50 Gil total. Once you are under that goal, take the cab back to Lindblum Castle. Return to Cid's room (Upper Level) and watch the scene there. After it's over and you step out of the conference room, you should have an ATE available to watch, called No Free Lunch.
Make sure you have under 50 Gil and return to the Business District once more. Head to the second screen, where the pickle lady is, for a scene. This isn't the ATE yet, but Zidane will hand over the money he has and Quina will rejoin your party. Double back to Lindblum Castle and take the elevator to the Base Level, where you want to ride the trolley there to the Serpent's Gate. Follow the path here to an outside dock area, where you can board the ship you will be using, called the Blue Narciss. Upon agreeing to board it, you should trigger an automatic ATE called Give Me My Money!, which is the one with the pickle lady.
That'll wrap up this visit to Lindblum, with your next destination being the Black Mage Village once more. However, since you have a means of travel, of course there will be some side questing you can do right now, so the next page will deal with that. If you wish to continue the story, then you can go straight to The Desert Palace (1) page.
Important: At this point in time, Dagger will periodically have "trouble concentrating" during battles, meaning she will essentially use up her turn doing nothing at all. Keep this in mind as you assign roles to characters in the upcoming areas.