Croatia - Dubrovnik

What are you interested in?
Roman amphitheatre: Pula,
Roman palace and ruins: Split
Medieval towns: Dubrovnik, Korcula, Hvar, Split, Trogir, Sibenik, Rab, Rovinj, Porec
Sandy beaches: Lopud, Lumbarda (Korcula), Partizanka (Hvar), Lopar (Rab) and others: www.croatia-beaches.com/sandy
National parks, waterfalls: Plitvice, Krka (near Sibenik)
"Venitian" architecture: the whole coast
"Austro-Hungarian" architecture: Zagreb and the small towns in Slavonia (close to the Hungarian border)

By far the most relaxing way to visit Croatia are trips by ferry. There are two weekly ferries going down from Rijeka up to Split - Dubrovnik - Bari (Italy). Between Split and Bari, they run almost daily (3 different companies). There are also daily ferry crossings from Ancona and Pescara in Italy to Split and some Islands.

The most popular Islands are Korcula, Hvar and Rab, where tourists mainly concentrate in the "capitals" with the same name. Less frequented places are Lastovo, Mljet, Brac (except Bol), Vis, Pag, Susak, Ilovik, etc.

Have a look at the following sites: croatiatraveller.com, dubrovnik-online.com, korculainfo.com, lastovotravel.com, lopud.eu, jadrolinija.hr,
krilo.hr/raspored_en.htm, lariveralines.com, losinjplov.hr/veze.htm

Dubrovnik is very expensive to stay whole time. I'd suggest to stay perhaps in Cavtat which is 15 km on south and much cheaper.

2-3 weeks is a long period, so, if you are interested in this part of croatia than I can suggest:

konavle - in the hills above DU and Cavtat with specific folklore and many village restaurants
peljesac - half island about 50 km north of DU - places ston and mali ston with famous restaurants (mussels), vineyards near Potomje and Postup, famous windsurfing site Viganj, beaches in trsteno and zuljana ...
river neretva - between Metković and Ploče - boat swamp safari, eels and frogs are speciality there.
elafiti - small group of islands just near dubrovnik - Lopud, Sipan, Kolocep with nice sandy beaches and beautifull vegetation
trsteno - arboretum 10-15 km north of DU
mljet - island and national park with boat connection to DU
boka kotorska in montenegro, fjord with beautifull views and city of Kotor
... etc

It's good to know that Peljesac (Trpanj) and Neretva (Ploce) are connected with ferry.

I visited Split, four hours away on an ultra modern bus, and I loved the city. Buses run up and down the coast between these two cities non stop every day, they are inexpensive, and the scenery is simply stunning the entire way. Dubrovnik is the greatest too...

If you have 2-3 weeks in Croatia you can see both cities, and many of the islands.

It is a great country, and the time will be enjoyable.

Taxi fare from airport to Dubrovnik centre is about €25 to the Old Town - to Lapad area where lot of hotels are will be more.
With 2 or 3 weeks, I recommend leaving Dubrovnik for a while and going up further south to spilt or beyond and also out to the islands as others suggested. There are many that cane be visited from Dubrovnik on daytrips but with that length of time you woudl be better maybe travelling by bus to Spilt, and then making your way back to Dub by ferry stopping on those island mentioned. You will have more than enough time to do this and in a relaxed way in 2/3 weeks. I did it in 2 and it was fantastic holiday.

As an alternative to taxis, there is a bus from the airport that makes 2 stops in Dubrovnik - one just outside the gates of the "old town", and another in the newer part of town. It was cheap, much cheaper than a taxi. I recommend a side trip down to Montenegro - I just took a bus and did a day trip to Herceg Novi - about an hour south of Dubrovnik - but wish I could have seen more. If you go to Split (I highly recommend it), check out a performance at the National Theater.