The Old City is best explored on foot. For longer distances across modern Damascus, minibuses are the cheapest option. They hold about 12 people and run fixed routes, though the destination is written in Arabic on the front, so you may need local help.
Yellow taxis are plentiful and a good alternative. Insist the driver uses the meter. A ride from the airport to the city centre costs around $17; rates go up at night. Private cars can also be hired for inter-city trips, typically $10-25 depending on the destination.
Damascus has two railway stations, though trains are slow and schedules unreliable for getting around the city. For leaving Damascus by bus, private air-conditioned coaches serve other Syrian cities, but schedules change frequently, so book ahead when you can.