We will be in the Split area for 4 nights, with a car. . If we also want to go to several islands, should we consider staying on one of the islands for 2 nights or just using Split as our base for all 4 nights?
It depends what you want to visit from Split as a base. There isn't so much to see in Split alone compared to other parts of Dalmatia. Couple of hundreds of meters of harbour with cafes and palm trees near Diocletians palace and that's it, the rest is more or less like any other bigger city. Makarska eg is much nicer place for stay. However as a base for visiting other places Split is good. There are direct ferries from Split harbour to Hvar and Brač so no need to stay on islands and limit your mobility. From Split you can visit National parks Plitvice or Krka, Šibenik and Zadar on north, and Makarska, Dubrovnik on south. The most bautiful for me personmaly is Makarska riviera and very scenic road with a view to several small beautiful towns on the Adriatic coast (Brela, Makarska, Tučepi, Drvenik, Igrane etc). So if you are not going to visit Plitvice or Zadar, maybe your best option would be to stay in Split 2 days including visit to eg Hvar, and other 2 days in Makarska or even Dubrovnik.
We spent 4 nights in Split and loved it. It's a beautiful city with different architectural styles. We wandered around the city on our own, visited the market, and then we had a private guided tour of Old Town and Diocletian's Palace with Maja Benzon, who is recommended by Rick. The private tour was awesome! Maja is warm, friendly, and very knowledgeable. We also enjoyed the Ethnographic Museum. There are other small museums you can visit if you are interested. The Mestrovic Gallery seems interesting but we didn't have time to go there. Split merits more time than just a day but that depends on what your interests are.
We took a day trip to Hvar Town via catamaran. That was one of the highlights of our trip. Hvar Town is gorgeous! What time of year will you be there? We were there in September, and it was not party central at that time. I understand there are lots of party goers during the summer months. If you are going in the summer, I understand that Stari Grad on Hvar is quieter. Whether or not you stay overnight on Hvar is up to you. I'm sure it is a lovely place to stay overnight. But that depends on how much you want to change hotels. A day trip is so easy. We took an early morning catamaran (around 9AM) and we returned on either a 6PM or 7PM catamaran. The trip is about an hour to Hvar Town.
We visited Trogir another day. That was about a 3/4 day trip. Well worth it. Other day trip possibilities from Split are Krka National Park and Sibenik.