To get to Cardamena from the airport you can take a shuttle bus (ticket price is about 3 EUR). The schedule and prices can be found here.
If you'd like to visit the island's capital before heading to Kardamena, you can take a bus from the airport to Kos city center (the bus stop is located at the terminal exit). Buses run from 7:55 to 19:40. You can also take a bus from the capital to Kardamena (the stop is located at the bus station, 7 Cleopatra Street). Buses run from 7:30 to 21:00. The interval of movement is 1-2 hours. The price is about 7 EUR.
But the most convenient way to get to Kardamena from the airport is by cab or shuttle or mini-bus. The cab stand is located at the exit of the terminal. Cab price is about 35-40 EUR, transfer price - from 12 EUR. Luggage is usually paid separately. For more details about the transfer, see here.
