I.K.A. (Greek NHS) doctors can be found in larger towns and some villages, surgery hours are normally 08:00 - 13:00, Monday - Friday, although they may not be present every day as they may to have to cover several surgeries.
In the larger towns and tourist resorts there's likely to be private practices and, in tourist resorts, these are likely to be open much of the day and maybe Saturdays (during the tourist season). Either way, expect to pay for your consultations and prescriptions.
Health Insurance Card (E111 replacement) covers treatment for major
injuries / illness but is no substitute
for adequate travel insurance (and
doesn't cover, e.g. repatriation).
Private doctors can usually accept payment by credit/debit card and may
be able to liaise directly with your travel insurance company, if needs
be. If you have to pay and then reclaim, be sure to keep receipts for
all your expenses, including prescriptions and taxi fares, etc. I.K.A.
doctors usually have no facility to accept credit cards or medical
insurance so will need to be paid in cash, getting a receipt may be a
problem. In the event of minor ailments and injuries it's often worth
visiting the pharmacy before
seeing the doctor as pharmacists are able to dispense a wider variety
of medicines than in the U.K.
On Kefalonia you should be able to consult a doctor in: Argostoli, Lixouri, Skala, Poros, Sami and, possibly, Agia Efimia and Fiskardo. Pharmacies can be found in Argostoli, Lixouri, Poros, Skala, Sami and Vlakhata (near Lourdas). The Greek word for doctor, ΙΑΤΡΟΣ (pronounced 'yatros'), and a red cross, will normally be displayed outside the surgery / medical centre.
In the event of serious illness / accident, or when the doctor / medical centre is closed, there is an I.K.A. ('N.H.S.') hospital in Argostoli which has a 24 hour, 7 day, emergency department where you will be treated for a very small fee (three - five euros). In serious cases they may require your medical insurance details or European Health Insurance Card (this can usually be sorted out later). Although there are a couple of ambulances on the island they will usually only respond to call outs from a doctor or the police and it's often easier and quicker to make your own way to the hospital, e.g. by taxi.
Emergency dental treatment can be obtained during the morning at the hospital in Argostoli or at private dental practices (e.g. in Argostoli or Poros), normally during the hours of 08:00 - 14:00, Monday - Friday. Some dentists are open evenings and may be able to make a special appointment in the event of severe pain.