From here, take a short trip with Olympic Airways to Samos.
In summertime, there are 1-2 ships from and to Kusadasi/Turkey as well!
You can take a flight to Izmir, hop by bus or taxi to Kusadasi and take the ferry.
They leave around 9am and 5 pm from Samos and Kusadasi respectively.
You will find a flight from near your home in our flights section
There's still a way... you can reach the Greek port of Patras by ferry from the italian ports of Venice, Ancona (both easy to reach by train and car), or Bari and Brindisi (both down in the south of italy but the boat trip is shorter and cheaper).
From Patras, you can take a bus or a train to Pireas (many ferry companies offer cheap or even free bus shuttle service to Athens!).
A quite comprehensive list of ferry companies serving these routes is here
In Pireas, enter a ferry boat to Samos, there's at least one going every day, in summertime 2-3. Time varies between 8 and 17 hours.
The most accurate timetable is GTP.gr
If you're coming from another part of greece, or if you plan some island hopping:
There are also a lot of connections by boat with a lot of islands (f.e. Kos, Rhodos, Mykonos, Paros, Naxos, Chios, Lesvos, Limnos, Ikaria and many others) as well as the northern towns of Kavala and Thessaloniki.
Unless you are planning to tour the mainland as well and/or plan to stay longer it does not make sense.
The ferries charge sometimes more than 150 Euros for a car, and that's one way!
For this kind of money you can rent a car for a while already.
That said, it can be fun to travel to Greece by car if you take some time.
The cheapest (and interesting) way nowadays is travelling through the former Yugoslavia (Croatia, Serbia, Macedonia/FYROM).
It's perfectly safe, hotels on the way and restaurants are good and very cheap and if you plan your trip right you can see some beautiful cities - such as Zagreb, Belgrade or Nis.
You will arrive in Northern Greece, and with some planning you can catch a boat to Samos right from Thessaloniki which is less than 100 kms from the border.
They don't sail every day though, make sure you check the timetable.
The alternative is driving to Italy and take a ship (see above), which means a good 1000kms less to drive but also some hundreds of Euros more to pay if you want a cabin.
In the end it's up to your liking, either way the trip will take around 3 days from central Europe.