If you have the privilege of coming in a private boat the island is a perfect destination but even without it you can get to lots of remote beaches with crystal clear water and trees shading you while you are enjoying superb swimming in the green-blue sea.
If you don't come in your car you will have to rent one or a motorbike to enjoy the beauty of the island. There is only one bus that runs along the only road along the island from Vathy to Kioni at the far end.
Big ferry boats come and go daily from Patras, Lefkada and Sami in Kefalonia. Some run only in the summertime but the main boats from Patras and Kefalonia go all year round.
There is also a ferry connection with Italy.
Ithaki port police: +030-26740 32909. Sami port police: +030-26740 22031.
This early morning ferry to Patras is my favourite (there is another one at 16:00), as you can have from it a wonderful view of the island's coasts at sunrise. You have to get up early, OK, but the show is really (for me) worth the effort. The ferry stops at Sami (Cefallonia) and then goes directly to Patras, where you arrive around 10am.
Ithaki is a rather small island with not many mountains, so renting a scooter is a good way to know the island around. There are a few shops renting bikes/motorbikes around the main square.
Some are strict and ask you for a valid driving license, some are not and don't ask... you choose!
There are 2 boats in the harbour offering trips to the beach of Gidaki. This beach is in the east south coast of the island and can only be reached by boat. The boat takes you there (at 10 or at 11am) in 10 minutes and picks you up at 14 or at 15h, so you can spend the morning there.
7 euros for the ride.
There are daily ferries from Patras, and from Kefallonia (Sami). BUT the last one from Patras is at noon, and from Kefallonia at 15:00, so you have to sleep there, because the next ferry back is at 7:00.
The ferry from Sami takes 1 hour.