I drove this exact route (plus to Mostar) a few years ago, without a GPS. I did not need one. I had looked up directions ahead of time (Google Maps) so knew where I was going. I had a detailed map with me too but barely used it.
The roads are well marked. Between Zagreb and Dubrovnik there is a modern freeway, the A1, most of the way (it doesn't go all the way to Dubrovnik yet I believe). To go to Plitvice, you detour off to a local highway for quite a while, but it still easy to follow if you figure out the directions ahead of time. The road signs pointing to the big towns on your route are well marked. There are probably signs off the A1 to Plitvice - I don't recall (a detour at -Karlovac I believe) but certainly there are for Split, Makarska, and Dubrovnik.
One way a GPS might help you would be in the bigger cities (Zagreb, Split, Dubrovnik). I had no trouble driving out of Zagreb, and I didn't drive into Split (came back later via catamaran), but I found driving into Dubrovnik a bit intimidating. I had trouble finding a petrol station to fill the car up before returning it there as well. But for driving between cities, a GPS is not necessary if you do a little research ahead of time. Even in the cities, if you use Google Maps, you can probably find where you are going without a GPS.