When entering Greece, EU citizens are not required to carry their passports. An EU Identification Card provides adequate information. UK citizens need to carry a passport, but otherwise the entry conditions are the same as other EU nationals.

Citizens of the U.S., Canada, Japan, Australia, New Zealand, Israel, Norway, Switzerland, Iceland and several South American nations require only a valid passport to stay in Greece for up to 90 days without a visa. Nationals from a number of other countries are permitted to remain in Greece for 60 days without a visa. Visa extensions can be easily acquired by an application and payment of an extension fee to the proper authorities, while you must also be able to show bank documentation or other evidence which states that you have the means to support yourself without employment. Staying in the country with an expired visa is prohibited and subject to prosecution.
 
Lastly, Greek authorities prohibit entry to individuals whose passport states that they have visited the occupied region of Northern Cyprus after November 1983. 

At the airport the customs area has three lanes. You should select the appropriate lane depending on the items you have to declare. If you have nothing to declare, you simply pass through the Green lane. In case you wish to declare business goods or the like, you should go through the Red lane. If you are an EU national, you can easily go through the Blue lane.

In order to bring your pet (dog or cat), it is essential that you present a certificate stating that the animal has no infectious diseases and that it has had all necessary inoculations during the last 6 days prior to arrival. It must also state that the animal has had a rabies shot in the last 12 months (for dogs) and in the last 6 months (for cats) prior to arrival.

The owners of dogs must also carry documentation certifying that the animal has been wormed against echinococcus. Certificates must be issued by a veterinary authority and authenticated and notarized by a corresponding medical authority and by a Greek consulate. 

Travelers will not face any major health risks in Greece. During the summertime though, visitors should be prepared to deal with the Greek climate. Be sure to wear a hat and sunglasses, and to apply sun-block while touring and sunbathing. Wear light and airy clothes and always have a bottle of water handy.

If any health problems do arise during your stay in Greece you will not have any difficulty in dealing with the situation. For minor injuries and illnesses, just go to a pharmacy, where they will be able to assist you immediately. Pharmacies can dispense most over the counter drugs. For more serious conditions, you should go to a Health Center or a Hospital. Health care in Greece is free and you will not be charged for treatment. Hospital and clinic facilities are first rate and the medical personnel is efficient.