I can help you classify your company staffing in QuickBooks Desktop payroll, @klm4mchr.
To properly handle your staffing company's tax processes, I recommend contacting your tax agency and tax expert. This way, they can help you with an accurate classification for your taxes.
You can also refer to the information given by droybal on how payroll taxes should be handled.
Additionally, you can generate payroll reports to gain a more detailed insight into your business's financial status. For more information, please refer to this article: Run payroll reports.
Should you have additional concerns related to payroll, feel free to post them here in the Community space.