Although there are ferry connections to Ithaka from Patras (Peloponnese), Astakos (on the inland opposite Ithaka), and Kefalonia (Fiscardo), I'd strongly recommend that you approach Ithaka with a rented car from Argostoli (Kefalonia).The boat connecting Kefalonia to Ithaka (its little port shows up as Pissaetos on websites) sails not from Argostoli but from Sami, the port on the east coast of Kefalonia; to make a bus connection with that boat, and then to take a taxi from the tiny isolated port where you disembark on Ithaka, costs far too much time. Rather, in your rented car, drive the 40 minutes from Argostoli to Sami; the boat fare for the car is 12€, for each individual 3€.
More information for ferries call Ithaki Port master office at phone : 2674032909
Once on Ithaka, you can drive to Vathy, the main town, in about 10 minutes, and you'll have wheels with which to explore Ithaka. Vathy itself is a little port, a miniversion of bigger Greek ports with their bustling tourist-oriented facilities.
You might enjoy a cold drink or coffee and admire the bay stretching before you, but otherwise there's not much to do or see here. Instead, drive 16km (10 miles) north to Moni Katheron; the 17th-century monastery itself is nothing special, but the bell tower offers a spectacular view over much of Ithaka. For a more ambitious drive, head north via the village of Anogi, stopping in its town square to view the little church with centuries-old frescoes and the Venetian bell tower opposite it.
Proceed on via Stavros and then down to the northeast coast to Frikes, a small fishing village. Finally, take a winding road along the coast to Kioni, arranged like an amphitheater around its harbor; this is the place to stay if you want to give some time to Ithaka. As for the sites associated with the Odyssey, what little there is to be seen is questioned by many scholars, but that shouldn't stop you; after all, it's your imagination that makes the Homeric world come alive. From the outskirts of Vathy, you'll see signs for the four principal sites. Three kilometers (11?2 miles) northwest of Vathy is the so-called Cave of the Nymphs, where Odysseus is said to have hidden the Phaeacians' gifts after he had been brought back ( to the little Bay of Dexia, north of the cave). Known locally as Marmarospilia, the small cave is about a half-hour's climb up a slope and is near to Eumaeus Villas.You can visit it during hiking.
The Fountain of Arethusa, where Eumaios is said to have watered his swine, is about 7km (4 miles) south of Vathy; it is known today as the spring of Perapigadi. The Bay of Ayios Andreas, below, is claimed to be the spot where Odysseus landed in order to evade Penelope's suitors. To get to the fountain, drive the first 3km (2 miles) by following the sign posted road to the south of Vathy as far as it goes; continue on foot another 3km (2 miles) along the path. About 8km (5 miles) west of Vathy is the site of Alalkomenai, claimed by Schliemann among others to be the site of Odysseus's capital; in fact, the remains date from several centuries later than the official dates of the Trojan War. Finally, a road out of Stavros leads down to the Bay of Polis, again claimed by some as the port of Odysseus's capital; in the nearby cave of Louizou, an ancient pottery shard was found with the inscription "my vow to Odysseus," but its age suggests that this was the site of a hero-cult.