You'll probaly arrive in Bari via the Port or the train station. See the comments below on how to get from one to the other.
Train to Ferry: Exit the train station and across the street you should see a bunch of buses. Buy a bus ticket from the yellow huts, or from a newstand in the station, and get on the 20/ (yes, the / belongs there) but be sure to ask the driver for 'porto'. Validate your ticket in the yellow box on board. Once at the port you may be required to show your passport so have it ready on the bus. Next the bus should (ours didn't) drop you at the terminal of whatever ferry you're sailing with [I recommend reserving tickets ahead of time via web]. Once you've checked in or bought your ticket there should be a small shuttle van to transport you and your luggage from the terminal to the boat itself. Bon Voyage!!
Ferry to Train: I only went the other direction but I'm assuming it's the same procedure backwards. You may have to ask at the terminal when you get off the boat where to catch the bus... [if you've gone from port to train in Bari, let me know]
The railway station is THE hub for Southern Apulia. The Intercity train is roughly 8 hours from Bologna and reservations are strongly recommended as the long distance fast trains are usually jammed. DO NOT EVEN CONSIDER driving to Bari. You may be able to handle the driving habits of the Southern Italians, but you won't find any place to park and a rental car is a prime target for theft and break ins.
Bari is easily walkable if you can tolerate the traffic, and is a good base for daytrips. The fascinating city of Matera is about 90 minutes away by train, and Lecce is a couple of hours to the South.
Bari can be reached from the airport as follows:
"Autoservizi Tempesta" http://www.autoservizitempesta.it/index2.html
offers a shuttle service connecting Bari-Palese airport to Bari railway station.
bus no. 16 to the centre (Piazza Moro)
Radio Taxi tel. no, +39 080-5543333
Since I'm coming from East (Greece), I took a ferry via Corfu Island (Greece) and landed to Brindisi.
PHOTO : DE TORRES FOUNTAIN