Some means of (preferably motorised) transport is a must if you want to see a bit more than just your hotel, the main road and the town beach.
There are about a handful places in Kefalos where you can rent scooters, dune buggies, ATVs, and cars: Manolis, Trust, Katerina's are the most prominent ones. All of them are on the main road within walking distance, so you can easily compare rates and models. A simple and fairly modern ATV ("quad") suitable for 2 persons can be hired for about 15 € a day during the main season (August), for instance.
You are however well advised to check the insurance terms - full motor hull insurance is not always included, tyre damage is (as far as we could tell) never covered.
Here is a Kos Road Map for you take get an idea of the roads before you arrive.
It was the free map that we got with our car hire, but I can't get the whole map on one scan, so this is the Kefalos side, together with most of the Island, but excluding Kos town.
Hired a Suzuki Jimney from them for 2 days from their site near the Oasis Supermarket in Kefalos (they also have a branch nearer the harbour). We wanted a Samurai, but they didn’t recommend this with young children as the rear seats are too high relative to the sides. They also said the Jimney has power steering and ABS. Seemed genuine bunches who are a family run firm, operating for many years. They charged us October rates in September, which worked out at 80 Euros all in (included extra personal insurance at 2 Euros a day which they said was compulsory with children, and 2 Euros per day for an additional driver). The deposit was just 20 Euros. They provided a complimentary map which includes suggestions of places to visit, and they told us where we could find free parking in Kos centre. We collected the car at 9am and could either return it at 9pm on the second day, or between 8 and 8:30am the day after. We would recommend them, but there is a 200 Euro excess, and all damage to tyres is the responsibility of the customer - a no fault puncture could turn out to be very expensive. It’s worth noting that the conditions of hire state that the driver must not consume any alcohol. Also, check the car carefully for damage when you sign the agreement; ours said there was no damage but we later noticed the light lens was cracked – fortunately, they did not spot this or make an issue of it when we handed it back.