If you decide to go to Dubrovnik by car, you must have a valid driver's license and car insurance. On the way from Split, you will have to pass the border to Bosnia & Herzegovina, so please check if you need a visa for Bosnia & Herzegovina. If you are coming from Zagreb, take the highway to Ploče, then continue to Dubrovnik on the Adriatic Road along the coast. Driving from Zagreb to Dubrovnik takes about 6 - 8 hours. Keep in mind that traffic is heavier during summer months and that border queues are possible at border crossings.
If you are coming from Italy, there are several ferry lines from Ancona to Zadar and Split or from Bari to Dubrovnik. For more information please check www.jadrolinija.hr.
The Croatian Automobile Club website - www.hak.hr provides detailed information on road conditions and useful information about routes.