You don't have to follow a legal process to start using a new name but it would be necessary in order to change official documents like your passport or driving licence.
You can legally change your name, or that of your child, by Deed Poll and this is conducted through the Royal Court.
If you do change your name for any reason, you will need to replace your passport for one issued in your new name. You may also need to apply for a new Resident Certificate/Employment Permit (Right to Work), which you can do here.
Soon you will be able to register your new details with various States' departments all through one online form, but for the time being you will need to contact them individually.
