public transportation on Islands

That's a very important, and intelligent, question to ask if you don't intend to rent a car. The BusCroatia website is a pretty good source of info on Croatian buses but I came up fairly empty for Hvar and Korcula except for multiple daily buses between Hvar Town and Stari Grad. It's possible the website doesn't have data loaded for the peak travel season next year, when there may be more bus service.

I think your best bet is probably to start with a comprehensive guidebook. You need one that covers most if not all of the islands. Even an older edition will probably mention the existence of specific island bus routes and the names of the companies running them. That would help you identify the likeliest islands and allow you to Google for specific and more current information.

You can also Google "Buses on Korcula", etc. I just tried that and came up with this website and a link to a schedule for what seems to be the only bus line on that island.

I only visited Brac and did use a bus to get around the island (from Supetar to Bol). Another option to consider (which may be cheaper than a car) is a scooter. Traffic was nonexistent when I went (early September). Relying on bus service really depends on where you're trying to go and how flexible you are. The frequency may not be great during some parts of the day, and you may have to plan around the bus schedule or you won't be able to get back that day. I caught the last bus from Bol to Supetar, I recall. It's doable but you just need to work with the bus schedule.

The bus services on most islands are thin on the ground in my experience and to explore, you should hire a scooter or a car. Taxis there are expensive. If you are staying in one place for several days, consider staying in a larger resort such as Korcula town or Stari Grad or Jelsa on Hvar. Vis is an island that IMO requires a car.

We just had 4 nights on Vis and were happy with public transportation. One day we took the bus to the pretty little village of Komiza. Another day we took a boat trip around the island, including the nearby Blue Cave. We really liked Vis ... Enjoyed sitting by the harbor watching the yachts come and go. Some of them are very entertaining. Vis is easily reached by ferry from Split, and the early morning return ferry gets you to Split in time to catch the bus to Dubrovnik.

We didn't have a car on Korcula but did take an inexpensive taxi to a beach one day. On Hvar, we just took boat trips to the tiny nearby islands (i.e. Stepanska).

Other thoughts on Croatia: I would highly recommend you check the cruise ship schedules before planning any visit to Dubrovnik. You will want to avoid being there when thousands of cruisers are in town. And I would recommend staying in the little nearby village of Cavtat instead of Dubrovnik. You can easily visit Dubrovnik via boat or bus but can escape back to gorgeous Cavtat at the end of the day.

Also highly recommend a day trip to Montenegro, which you can easily do from either Dubrovnik or Cavtat.

We have been going to several Croatian islands for years. We used to rent cars, which helps, but you can get to all the main towns on the islands by a bus. I use the Autotrans.hr website for the major routes, (click on the UK flag icon for English)and it also includes other companies' buses. Given that villages need buses to stay functional, many have at least one a day. Remote Orlec, Cres,, for example.
Long distance buses now cross Europe in comfort. I used a Deutsche Bahn bus from Zagreb to Munich to get around train cancellations due to that year's refugee influx. This year I crossed Croatia from Rovinj to Zagreb and down to Veli Losinj on Losinj island by bus. A bus ticket must be purchased with a seat reservation (sjedalo broj 23, for example) to avoid being kicked out of your seat. You may need to buy the seat reservation coupon at your destination for the return trip, not in Zagreb. Look up your bus trip online, write it on a piece of paper, and hand it to the bus depot clerk for accurate ticket purchase. Many depots are busy and loud.
If you must see Dubrovnik, consider staying over on the Elafiti islands which are a water taxi or ferry ride away and spectacular.
By staying in any town for several days, you will see the real Croatia, not just the surface. Great plan. Walk around, enjoy it all. You can rent a bike on almost any island, for several days if you want. Car rental is a headache you can avoid,

It is a harder question to answer than you would think, but overall, any town you would want to spend several days in will have fairly good options *(as well as enough to captivate you near the town). If not bus travel, then day boats and the like are options, as mentioned above. Most islands have one main town and you may not have much need to explore the other small villages. Perhaps you could tell us what places may have stood out to you in your initial reading and we can offer more specifics.

One of the oldest backpacker route is: Split - Hvar - Korcula - Dubrovnik. You don't need the car. Just catamarans. Catamarans will take you into the heart of those small coastal towns. In case you want to visit national park Plitvice, than with the car is easier. Sure you can use busses and trains. Check first does hostel/hotel have transfer to entrance in case is far from the entrance.

Have a good time!