Travelling around Croatia, just like travelling to Croatia, can be tricky and it requires some planning. Zagreb is Croatian main transport hub, and that's probably where you'll end up many times trying to go from one Croatian destination to the other. Keep in mind that everything in Croatia is seasonal, and the same goes to your transport options for travelling around Croatia. Transport schedules are more frequent during the summer months than the rest of the year. The same goes for fares. They increase during the summer months.
Travelling around Croatia: modes of transport
Travelling around Croatia by bus is very convenient and relatively cheap. All Croatian regions, from east to west, and north to south are well connected by buses. However, some areas, like Istria and Croatian islands, while easy to reach by bus, aren't so convenient to travel around on bus. I would highly recommend you to have a car should you wish to explore Istria or Croatian islands. Here, buses are infrequent and in many cases very limited. Croatian buses are modern and comfortable.
Travelling around Croatia by train is adventurous, to say at least. Train transportation is very limited, and many towns (like Dubrovnik) aren't even connected by train. And the trains are slow here. However, if you are visiting the northern and eastern Croatia, train transportation can be a decent transport option.
Croatia is relatively small country, and travelling around Croatia by plane often takes as much time as travelling by car. Only Split and Dubrovnik have regular flights to Zagreb. Reaching the rest of the country is easier and more convenient by bus, or by car. Zadar, Pula, and Rijeka are connected with Zagreb and Dubrovnik by plane during the summer months. The major national carrier is Croatia Airlines.
Travelling around Croatia by ferry is doable, and not that difficult as it might seem at first. Ferries in Croatia are reliable and frequent, except for some far-flung islands. However, it takes a bit of effort to plan your island hopping trip in Croatia. The main ferry operator is Jadrolinija, and on some less frequented routes you'll find smaller operators. During the high season there are also number of catamarans connecting islands with mainland.
Travelling around Croatia by car is definitely the most convenient way to travel. Croatian roads are good. There is a new highway going south all the way to the town of Ploce, 90 km north of Dubrovnik. However, to reach Dubrovnik you'll need to drive this last 90 km on a coastal road. This road is a bit winding, but the drive is scenic and highly recommended. Highway also connects Zagreb with Rijeka to the west. And Istrian main road, called Istrian Y, is partly highway (double lanes), partly national road. Highway also connects Zagreb with Slavonia. Traffic on the Croatian roads is seasonal. But even during the high season, you can feel a bit of the traffic only from Zagreb to Zadar, and Zagreb to Rijeka. The rest is a smooth drive.