For this you would likely just create Ticket Custom Fields for those items (Setup -> Tickets -> Ticket Custom Fields). Those custom fields could be text boxes, radio buttons, drop down lists to choose from, or multiple choice boxes.
So you might have 3 or 4 custom fields:
First Name text field
Last Name text field
Start Date date field
Hiring Manager text field
You can set each one of those to be required if you want to make sure that it gets answered on ticket submission.
Once you create a ticket custom field and save it, an option will appear in the custom field settings to choose which Request Types to apply that to.
Here is an example where i have created a Request Type called "Software License Request" and then I have made some Ticket Custom Fields that i want the submitter to answer. The ones with red asterisks are the custom fields that i made 'required':