In manga, they met in episode 416-230. It was when the team was defeated singlehandedly by Bartolomew Kuma and they were sent in different locations. Luffy was sent to the island where Boa Hancock was the leader and became friends though Hancock have a secret love for Luffy.
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his grandpa is Monkey D. Garp
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408
Hancock and luffy's crew and shanks
Shanks loses his arm while saving a young Luffy from a Sea King. Luffy had just previously been abducted by a group of bandits, the leader of whom had taken Luffy out to sea and had been subsequently eaten by the same Sea King.
The fight between Luffy and Crocodile starts from episode 122
Episode 236
episode 37 the arlong arc