Extract/export ticket data from Zendesk to another ticket platform using the API

Hello,

We are trying to extract all tickets from our Zendesk instance and send to another ticketing platform. I understand that this probably has to be done using the API but I cannot find anything specifically to give to our developer due to the range of things we need from the tickets.

Things we need extracted from the tickets:
Subject
Requester info (name, email, org., requester type)
Assignee
Form
ALL ticket comments (internal, external, call logs)
Ticket events 
Attachments
Category and sub-category
Type
Jira ticket #
2-3 Custom fields

If someone could just point me in the right direction in the documentation please, I would greatly appreciate it.

Thanks.