If you are an EEA or Swiss citizen you can live, work and study in the UK without needing permission from the Home Office. The rights of your family members are more complicated.
Once you have exercised Treaty rights in the UK for five years you become ‘settled’ in the UK – that is, have a right of permanent residence. If you plan to apply to be naturalised as a British citizen you will need to apply for a Permanent Residence Card once you have completed five years of Treaty rights.