Athens has a good underground system, and Thessaloniki is in the process of constructing one, too (expected to open at the end of 2020). Note that only Greek student cards are valid for a student ticket on the metro.
Taxis are widely available in Greece, except on very small or remote islands. They are reasonably priced by European standards, especially if three or four people share costs. Many taxi drivers now have sat-nav systems in their cars, so finding a destination is a breeze as long as you have the exact address.
City cabs are metered, with rates doubling between midnight and 5am. Additional costs are charged for trips from an airport or a bus, port or train station, as well as for each piece of luggage over 10kg. Before you get into the taxi ask how much the price is likely to be. In some places, such as on islands, where they shuttle tourists between popular locations, the price is set each season.
Some taxi drivers in Athens have been known to overcharge unwary travellers. If you have a complaint about a taxi driver, take the cab number and report your complaint to the tourist police. Taxi drivers in other towns in Greece are, on the whole, friendly, helpful and honest.
Most Greek towns are small enough to get around on foot. All the major towns have local buses, but they are especially useful in Athens, Patra, Kalamata and Thessaloniki.