Hospitals
Newquay has a hospital but in terms of emergencies it is only a minor injuries unit. (NHS website). Other than that, the main hospital for the whole of Cornwall is the Royal Cornwall Hospital in Truro (but if you need to ask for it everyone calls it Treliske) (website)
Pharmacists
While the supermarkets can deal with most things you might need there are three Day Lewis Pharmacies in the town and a Boots on the main street
Garages (mechanic)
Vets