Flying to Bari is definitely the best option if you're trying to reach Dubrovnik or the surrounding area.

Bari - Dubrovnik by Ferry
Jadrolinija operates an overnight ferry on this route once a week during off-season, and several times a week during summer.

Bari - Zadar by Air
Ryanair will commence twice-a-week flights (on Tuesdays and Saturdays) from Bari to Zadar the week beginning 1st June 2010.

Flights from the UK to Bari
Ryanair have flights from London Stansted and Dublin
British Airways have flights from London Gatwick

Useful Travel Links
Bari Airport Includes information on getting to/from the airport
Trenitalia Train timetables for journeys within Italy