Once your are in Dubrovnik you should definitely take a Cable car ride to the top of Mount Srdj above the city. From there you can enjoy the amazing views of the whole city, the Elaphite islands and the Adriatic Sea. Cable car ride is a short and in only 3 minutes you will be at the top of the Srdj.
There are Panorama Bar & Restaurant if you wish to have a drink or dine while enjoying amazing views of the area. While there you can visit Homeland War Museum in the Fort Imperial to learn more about recent Dubrovnik's history.
Instead of taking a cable car to reach the top of Srdj Hill, consider booking this bus panorama tour by cabrio bus (6 departures throughout the day).
It is very easy to reach Dubrovnik Cable car from Old Town or Pile or Ploce Gates. From both gates you can reach it by following the pedestrian sidewalks (from Ploce Gate you'll need about 4-5 minutes while from Pile Gate it takes 8-10 minutes).
From Old Town you can do this by walking up the stairs from Stradun by taking Boskoviceva street. When you reach the top of the street you need to walk across parking lot towards stairs below cable car.
For visitors arriving from cruise port the easiest way to reach the Cable car is to take city bus no.8 which will drop you exactly at the spot or take a taxi from the port.
The Cable car tickets can be bought from the receptions of the many hotels in Dubrovnik, from several travel agencies in the city, at the Cable car stations (can be paid in KUNA or BY CREDIT CARD).