If your landlord serves you with a termination notice you can move out any time before the notice ends without having to give your own notice. If you were given notice because of the end of the fixed term, you are responsible for paying the rent until the last day of the fixed term. Otherwise no further rent is payable from the day you hand back vacant possession after getting a termination notice from the landlord.
If your landlord serves you with a termination notice and you do not vacate by the day requested, the next step they can take is to apply to the Consumer, Trader and Tenancy Tribunal for a termination order. The Tribunal will send you a notice to attend a hearing.