You can use Custom Fields to customize the Ticket Submission form. Custom Fields are linked with Custom Groups, which decide the location (User ticket submission form, Staff ticket submission form etc. ) of the Custom Fields.
You can add a Custom Group from: Admin CP -> Custom Fields -> Groups -> New: Location.
Next, you need to add Custom Fields from: Admin CP -> Custom Fields -> Fields -> New -> Select the Custom Field Group to associate with. You can also select the type of custom fields that you want to use (Text, Radio etc.)
You can click here for more detailed information on Custom Fields.