You can fly with Croatia Airlines (quite cheap - 'family' discounts for married couples!) or take the bus/coach to Dubrovnik.
Alternatively, if you want to see Split or to travel by boat from Split, then you can go by plane, bus or train to Split (or further up the coast) and then take a boat or bus to Dubrovnik from there. The trip takes longer than it looks on a map.
This map is one jumping off point . The hotels to the west by the beach are quite a long walk or a local bus ride away from the old town. Buses run frequently and until 2.30am apparently.
Most hotels and private accommodation below respond quickly to e-mail.
Dubrovnik-online is an excellent site for cheaper private accommodation among other things.
Another site is ljeto.
Another is dubrovnik-apartments.com.
Croatian Tourist Board
Dubrovnik - Neretva County Tourist Board
National Park Mljet - an incredible daytrip from Dubrovnik by boat (you can also stay overnight on the island, for example, at Hotel Odisej). It's a good idea to rent bicycles at the park. Note that there's no bridge at veliki most.