I always thought that the best way to know a city is to explore it using public transportation. It tells a lot about the way a community is organised, a reliable crash-test. Additionally, you meet locals on their way to work or to school or going out for fun or shopping, and you get the feeling of their everyday life.
In Thessaloniki things are rather easy to figure out: the city expands along the coastline so you either have to move to the East or to the West. Or take a short ride to the Ano Poli, that is to the North. Here is how you can do it.

afijeis oasth
Get on the bus…
If you choose to get the bus (OASTH), you can find all information (in English) about routes, stops, even optimal routes, at http://oasth.gr/. If you are lucky and there is this electronic sign at the bus stop, you will know exactly how long it will take until your bus arrives. If not, try http://m.oasth.gr/ on your smartphone. In this case you might need some help, because it is only in Greek. Ask any local, many Greeks speak English and almost all of them speak… the body language.
Make sure you buy your ticket at a kiosk or have the exact ticket price, because the machines on the bus do not give change!

ktel grammes 1
There is a bus line (No 78) that goes all the way from the airport to the city center, to the train station and ends at the intercity bus* terminal (KTEL). Lines 8, 12 and 31, from the eastern areas and the center of the city, will also take you to KTEL.
From there you can get the coach to most Greek cities, but also to some destinations abroad, as I recently found out. You can find all the information you need here: http://ktelmacedonia.gr/

 *To Chalkidiki. All coaches to Chalkidiki leave from another terminal at the eastern part of the city. At http://www.ktel-chalkidikis.gr/ you will find out how to get there by bus from the airport, the train station, the KTEL terminal or the city center.

taxi
… or get a TAXI
Taxi drivers is another favorite source of information for me. They know – and talk about – all kinds of things: where to go for god food, entertainment, shopping, clubbing, picture-taking, any question will be answered. Their favorite topic here in Greece: politics. Many Greek friends of mine have told me that taxi drivers are somehow the… barometer of election outcome!
If you need to make arrangements for a taxi drive in advance, here are some agencies:
TAXI Way (I think it is the largest in Thessaloniki) http://www.taxiway.gr/
TAXI Thess http://www.taxithes.gr/
Thessaloniki TAXI luxury transfers http://www.thessalonikitaxitransfers.com/
RadioTaxi Macedonia http://taximakedonia.gr/
RadioTAXI Mercedes http://www.radiotaximercedes.gr/
Make sure you ask about the extra charges: for transfer by appointment, for transfer to the airport or the train station and in some cases for each luggage.

Have a nice ride!