Getting around Dubrovnik is very easy using a public transport, in fact city buses run by Libertas. You can get from one side of the town to another in only 15 minutes.
13 bus lines (1A, 1B, 2, 2A, 3, 3A, 4, 5, 6, 7, 8, 9 and 17) operate within the city while regular bus service is available for the areas south and north of Dubrovnik (such as Cavtat, Srebreno, Plat, Zaton, Ston, Orebic, Brsecine etc). These lines are operated by modern air-conditioned orange buses that start running at about 5 a.m. and stops at around midnight.
Those travelling to other destinations within Croatia (Makarska, Split, Sibenik, Zadar, Plitvice, Zagreb etc) or abroad (Mostar, Sarajevo, Herceg Novi, Tivat, Budva etc) should head to Main bus station to catch one of many intercity buses.
If you are thinking about exploring areas south and north of Dubrovnik, consider hiring a car for a day (to visit Cavtat, Mlini, Trsteno, Ston and Peljesac peninsula).
There are two bus stops in Gruz area (the new port serving ferries and cruise ships): one just outside the main bus station (convenient for cruise passengers) and the other one next to the port building (opposite Hotel Petka) which is convenient for those arriving by ferry.
If you need to reach Old Town from here take one of the following city buses: no.1A, no.1B, no.3. Get off at the last stop - Pile Gate which is the western entrance to the Old Town.
To reach the Cable car embarkation station from the cruise port you will need to take city bus no.8 at Main bus station. The bus will drop you exactly at the spot (you will notice Cable car sign). Cable car embarkation station is located above the Old Town in the neighbourhood of Dubrovnik called Ploce.
If you are staying in one of many hotels in Babin Kuk or Lapad, take bus no.6 to reach the Old Town. There is also bus no.4 running from Hotel Dubrovnik Palace via Lapad to Old Town.