From the Port or Airport of Mytilini
Most people who book their trip to Eressos through an agency arrange to have a taxi transfer. Skala Eressos is almost two hours away from the airport or Mytilini town. Molyvos is about an hour. Vatera is under an hour and Plomari around half an hour. It is a beautiful trip and in a way even if you have rented a car it makes sense to take the taxi and get your car delivered to your hotel so you can enjoy the scenary, especially if you are getting off a long flight. You can contact any of the Lesvos Travel agencies about taxi transfers. If you are using an Athens agency they may not be as able to help you with this.
Rental cars can be delivered to the airport or the port. There is a bus terminal in Mytilini right by the small park on the waterfront (next to the government building and the theater). Backpackers can easily get from the ferry to the bus station. From the airport you have to take a taxi or the local bus. Buses all over the island are pretty frequent.