There are several ways to get to Thassos.

By Airplane

Thassos doesn't have an airport but there are two international airports which are nearby.

Thessaloniki airport is bigger, which means there are more flights available from many European cities. If you land in Thessaloniki you have 2 hours ride with bus or car until you reach Kavala where you can embark on a ferry or 2.5 hours to Keramoti where you also have a ferry with much shorter ride.

Kavala airport is smaller with mostly seasonal flights but from it you are just 15 minutes drive from ferry port in Keramoti and also, from Kavala sea port.

By Bus

This is the cheapest way to get to Thassos. There are multiple daily departures for Kavala. The sea port is near by Kavala bus station, and there you can board ferry to Thassos. You can also take connecting bus line to Keramoti but you will not gain anything in time or money by traveling to this port town closer to Thassos.
If you are coming from Thessaloniki pay attention for lines that are not direct. That way you will avoid many stops in smaller places on the way to Kavala.

By Rent a Car

If you arrive by plane you can rent a car from several agencies that have offices at the airport, or you can make the reservations over the internet and make arrangements with your agency to have the car waiting for you on the airport.
Driving to Thassos is quite easy to navigate and straightforward via highway that is in great condition.
Depending on your car preference, you can find deals as low as 15€ per day. Of course, this way you have additional costs of ferry ride for the car.

By Taxi or Shuttle Bus

Taxi is the most convenient but also the most expensive way to get to Thassos. You can hire one at the airport, or bus station if you are arriving to Thessaloniki that way.
Of course, if you arrive at Kavala airport taxi is the best option for you because the ride to Thassos is not very far and this way you avoid ferry cost for the car.
Shuttle bus is very much the same as taxi, which can be a good option for a larger group (5+) of people or family. There are many agencies that provide this kind of service, and you can arrange for a shuttle bus to wait for you at the airport.

Thassos Ferries

This is not optional - you have to take a ferry ride to Thassos :-)
Thassos ferry companies operate from Kavala and Keramoti, and have multiple departures per day, with Keramoti port having several departures more.
Keramoti is also faster, shorter and with cheaper lines. Lines from Keramoti disembark in Limenas, the capital.
Kavala is less convenient but can be good option in certain situations, especially if you wish to visit Kavala which is a good idea. Lines from Kavala disembark in Skala Prinos.