How many days would you spend in each city?

You can accomplish a great deal at Plitvice by arriving in the park mid- to late-afternoon, seeing some of the park then, and finishing up the next morning, going inside when the park opens. Although there are additional paths not included on the labeled routes, I think one night is the sweet spot for most visitors and wouldn't recommend 2 nights when you have only 10 or 11 days in total. Just don't try to drive in from Zagreb or Split and plan to see the park beginning at 10 or 11 AM. That will mean being surrounded by an unbelievable number of day-trippers.

The park has a new ticketing system that involves reserving a time-slot ahead of time.

I mentally write off my arrival day because sleep-deprivation and jetlag mean I rarely accomplish anything at all, so I wouldn't count that day as available for sightseeing in whatever turns out to be your first city.

Time in Zagreb should depend partly on your interest in art and decorative art museums. It has a lot of them. I especially recommend the small but wonderful Museum of Naive Art. In the not-art category, the Museum of Broken Relationships is popular and fun. Not interested in art? In that case one non-jetlagged day in Zagreb would be OK. Love art? You'd be hapy to have two non-jetlagged days and would need to plan carefully to fit everything in. The historic district is worth several hours of wandering.

If you like art, check the websites of the Zagreb Museums for days closed. If you don't have a guidebook that covers Zagreb well, consult Trip Advisor's Things To Do list to see what museums sound appealing.

There's a possibility the weather may break before the end of the yrip. If that happens, the south may remain pleasant a bit later than Zagreb. All other things being equal, I'd prefer to start in Zagreb. However, after consulting airfares and schedules you may find that things are not equal. And there's a good argument for going straight to Dubrovnik and getting over your jetlag there.

It's mostly buses in Croatia. I've been to all your destinations and have never rented a car in Europe.
.

I would fly to Dubrovnik and work my way back up to Zagreb. As much as we love Dubrovnik, i would spend 2 days there and add a day to Split. With 3 days in Split you can ferry to Hvar and back in the same day. (I haven’t checked the ferry schedule which might influenece your decision) or you can spend one day in Trogir. Short bus ride there and back. From Dubrovnik to Split is a bus ride. From Split to Plitvice is bus again. With the day not used in Hvar, i would add it to Zagreb. It’s hard to determine exact days because alot depends on your travel time. I always research travel options, length of travel time and departure times before I finalize an itinerary. When we are in Croatia we mostly rent a car or hire private drivers cause some buses and ferries only travel once a day or not when we need them. A bus from Rijeka to Dubrovnik is 12 hours, but driving it is only six. Once we flew, once we hired a driver, once we rented a car. This is where your research comes in and plays an important role. Planning is sometimes mundane, but necessay. Also, it doesn’t costs anything to get quotes from drivers. With only 10-11 days your time is valuable so i would check all options.

Both times I've been to Croatia I have rented a car (once from Dubrovnik to Zagreb) for the flexibility of choosing our arrival / departure times, especially to Plitvice. It's easy to drive in Croatia.

If you do decide to drive...

You don't need a car in Dubrovnik or Zagreb, so rent upon leaving your arrival city and upon arrival at your departure city.

You will also want to compare rental car prices. Checking a random 1 week period in September, Hertz had a 300 Euro dropoff fee from Dubrovnik to Zagreb. Sixt did not. Roughly similar pricing otherwise.

From a weather standpoint, you might prefer to start in Zagreb and work your way south to Dubrovnik. But practically speaking, flights might suggest your better option.

My ideal would be to fly into Zagreb, two nights. Bus to Plitvice, one night, bus to Split, time determined by your interest, catamaran to Hvar or Korcula, 2-3 nights, catamaran (bus in case of Korcula) to Dubrovnik, two nights.

Thank you all for such wonderful and thoughtful advice!
Do you have any suggested websites to look up bus schedule times or where I can find a private driver? Thank you again!

I've used BusCroatia in the past, but there may be better options. My usual advice in that part of Europe (though I think it's advisable everywhere) is not to fully trust any bus schedule information you did not get at the bus station itself. And even then...

Online info may be incomplete or wrong. I couldn't fail to notice in 2015 that if I stopped in at a tourist office to check on buses, they reached for the telephone and called the bus station at my departure point; they didn't turn to the Internet.

Buses to and from Plitvice seemed especially flaky. There are enough of them to/from Zagreb that it's not too much of an issue if you aren't dealing with a really tight itinerary, but if you're connecting Plitvice with Split, I think you need to be a bit more careful. I wouldn't plan to take the last bus of the day (or rather, what I think is the last bus of the day).

Hi Doug,
What is the name of the apartment you stayed at in Herceg Novi that cost $35 a night?
Thanks!