Many visitors coming to Kos take holiday packages, flight and accommodation but there is plenty of opportunity for independent travellers to get to the Island.
Cheap flights to Kos tend to be concentrated into the weeks from Easter until the arrival of winter. Kos Airport is in the centre of the Island. That said there are daily flights from Athens and summer flights from Thessaloniki. Ryanair does fly all year round from Frankfurt and Milan with a wider schedule in high season. Charter airlines increase the number of daily flights considerably.
There is bus and taxi transfer available from outside arrivals to your hotel.
The shortest ferry journey is across to Bodrum in Turkey but there are also ferries to neighbouring islands; Rhodes and Santorini are examples. A ferry comes from Piraeus on the Greek mainland while the Kalymnos ferry leaves from Mastichari, not Kos Town.
Considering that Kos is a fairly flat island, it is fun to bicycle through, although be aware of the rising daytime temperatures in the summertime. For the less energetic, perhaps rent a scooter.
There are taxis and buses for getting around as well as a road train from Kos Harbour to Asklepeion.