How to start the Lost Friends quest in Final Fantasy 7
In Chapter 3, you are introduced to the Sector 7 Slums, and as part of that, a number of characters who can help Cloud raise his profile as a mercenary.
All this is part of the main story, so you won't miss it. As well as your first Battle Intel Report, you'll be introduced to several side-quests.
These cover everything a fledging mercenary at the start of a role-playing game needs to spread their wings - from hunting rats in a dungeon to finding some lost cats. This is latter one is where the Lost Friends quest comes in.