The ship docks in Santa Cruz and the there is a free bus which takes you to the port entrance which is on the edge of the main town. If you are feeling adventurous, you can get a bus from the bus station to Puerto de la Cruz which is a lovely place to visit. The bus pass costs 12 euros for 2 people.
The port of Arrecife is a dump, however the bus to Puerto del Carmen or Costa Teguise can be caught on the main road. It takes about 30 minutes to get to Puerto del Carmen and about 10 minutes to get to Costa Teguise. The fares are very cheap.
The port for Rome is Civitavecchia which is one and a half hours away by excursion with the ship. The excursions are very expensive and this can be done on your own on the train for 9 euros return with a ticket called B.I.R.G. which also allows you to use trams, buses and the metro in Rome. The best station to get off at in Rome is San Pietro which is a 5 minute walk from Saint Peter's Square and there are regular return trains all day.
Although the port area looks initally grim, once you get into the main city it is absolutely beautiful, particularly the Avenue Garibaldi. It takes about 15 minutes to walk from the ship and there is some fabulous architecture.