You as an individual can register with the CIS.
This will enable you to work as a subcontractor and have your tax deducted at source at 20%.
If you don't register, and work as a subcontractor, you will have your tax deducted at 30%.
Registering as an employer with CIS, would entail first registering for PAYE, then using the information supplied from that department, to register for CIS.
Once registered for CIS, you will be able to employ subcontractors, and deduct their tax from what you pay them.
You will then have to submit monthly returns for CIS, and pay any tax/PAYE deducted to HMRC every 3 months, or inform them that you haven't deducted any tax/PAYE.
You will have to submit the return to CIS every month, irrespective of whether you have employed someone or not. You will be subject to a fine if you not submit the return on time (currently ÂŁ100), and another fine every subsequent calander month the return is not submitted.
You will then have to make an annual return for PAYE, along with your self assesment.
It is unlikely that you would want to emply the Scaffolding company as Subcontractors, as you would likely have to employ them for the duration of the time the Scaffolding is in place.