This page of the guide to Final Fantasy 7 Remake has a walkthrough for the Chocobo Search side mission available in Chapter 14 - In Search of Hope.
This is one of the longer side quests you'll find in FF7 Remake. You start it by talking to Damon. You can get to him by completing the Missing Children quest. As soon as you get close to him, the mission begins.
Head north towards Nuts 'n' Bolts Hills and Steel Mountain until you reach Lookout Point. Talk to Mirielle and then pick up the key item. It is on the ground, just behind the woman.
Then begin the Tomboy Bandit quest. Get to the church where you'll find Kyrie. After talking to her, you must go to the Colosseum to fight for her. This will allow you to complete the Tomboy Bandit quest and get the next item for the task.
The first cache is located at Steel Mountain, just north of Sector 5 Undercity. There, you earn Prayer Matter, Ruby Tiara key, and Moogle Medals.
You'll find another spot on the Collapsed Expressway. Right at the entrance, you'll find the first cache. You'll get Moogle Medals and Diamond tiara key.
To get to the cache, you have to re-enter the sewers. You're not going to do that if you don't continue with the story missions.
It's hard to miss that cache. Your companions will notice when you reach the area.
After the cache is opened, you have to fight Sahagin Prince. This is not a difficult opponent. Fire is his weakness. If you have such attacks don't spare them and you will quickly get rid of your opponent. The mission will be completed upon exiting the sewers.
Reward: extra skill points for Cloud. Another reward is finding a letter near Don Corneo's mansion.