arianaalex, where are you flying into/out of? What month? Do you prefer nature or sightseeing? Will you be renting a car or using public transport?
I like: 2 days Dubrovnik (could do an island day trip to Elafiti islands on your full day if you prefer, or just do a short hop over to Lokrum, or just get island time on Hvar and Korcula)
bus to Korcula (the 4 hr bus goes on a ferry at the end) for three days
ferry to Hvar for three days IF the ferry is running (why we need to know time of year)
Ferry to Split/Trogir for final 2 nights.
The above is of course if you are flying into Dubrovnik and out of Split, so when you fill in details let me know what you think of the above suggestions and I will adjust advice. I don't think you need a car the entire time, but I often suggest renting one while on an island such as Korcula, so you can get to the more secluded beaches and see scenery.