How-To: Using the Zendesk API to create tickets

We're looking to use the Zendesk API to create tickets from our own internal system. We've looked at the documentation, but we're not sure where to start. Can someone please provide a simple, step-by-step guide on how to use the API to create a ticket? We're particularly interested in how to authenticate with the API, and what the required fields are for creating a ticket. We're also curious to know if it's possible to set the requester, assignee, and custom fields via the API. Any help would be greatly appreciated!