You will almost certainly have to deal with the local town hall at some point. They vary slightly in the way they do things from place to place, but the underlying procedures are the same. You may go to the town hall itself or you can often transact business at sub offices nearer to where you live, particularly for the costal regions.
They will deal with the census (padron) information, and will issue Certificados de empadronamiento and Certificados de convivencia, opening licences for local traders, planning permissions, small works licences, “Vado permanente”, etc etc., In a seperate but often nearby office are The “Consorcio” who actually collect monies on behalf of most of the councils (not Adeje), and will deal with payments for Local taxes, Vehicle road tax, Local property tax ( Impuesto Sobre Bienes Inmuebles), Rubbish collection rates, Sewage rates, and in some cases Water rates, although many Ayuntamientos have a private company dealing with water supply, in which case they will be located elsewhere.
It is YOUR responsibility to make sure that the information supplied to the local authority is correct. Estate agents will often offer change of name of ownership with the local authority as a service, but do not assume that they have done this. In the same way DO NOT assume that when you move your name and indeed direct debit for these services will be stopped! As a rule of thumb the person who was in occupation as at January of the year in question is resposible for payments of local property tax. Make sure you are clear than your name and bank account have been removed from the local records if appropriate.