check this link http://www.jadrolinija.hr/default.aspx?lang=2
As You can see, there are just three options - Starigrad/Hvar, Korcula/Korcula and Sobra/Mljet
They all are very different:
Starigrad is an beautiful old town on island Hvar and in July is very crowded. If You like to stay there, book in advance. Search web about Hvar and you'll find many useful informations but, generally, this island is all but no quiet.
Same for Korcula - really very beautiful medieval town but also crowded. It's known as a birth place of Marco Polo.
On this island You can find smaller villages without so much crowd but it's not so easy to get there.
Sobra/Mljet - that will be my choice. Maybe not Sobra but half of this island is National park where You can enjoy in peace. It's not crowded but, also, is better to book in advance. There are many of private rooms and just 2-3 small hotels. All is available on web, usually with pictures etc.
Transport on islands can be a problem, specially on Mljet (it's small island with less than 1000 residents). Usually, buses are waiting for the boat but they are connecting just main places, so if Your room isn't on main road you'll need to arrange with your hosts to pickup you somewhere. For instance, just in the national park is one beautiful small village "babine kuće" but bus stops 2-3 km away (as I remember).
Just pay attention - not all of boats are going to Sobra. I find one at Jul 05 - be careful.
I hope it helps - enjoy Croatia.
