4 days in Amalfi coast and 4 in sicily

I would stay in Amalfi or Atrani, which is just a short walk away. Much depends on what else you wish to see while in the area, but with no other parameters than location, that is what I would go with.
Sicily is a large island, and four days really only gives you time to see 1-2 places. Why did you choose Sicily? Are you flying into Palermo or Catania? Are you flying in from overseas or ??? Knowing your overall plan will help others advise, but I would consider sticking to the mainland to see more/travel less.

What do you consider budget, and what do you need in an accommodation, such as AC, a lift, breakfast, etc? How many people? That would help for recommendations. Also do your research on booking.com, but book directly for the best price. Share travels!

Thanks to your replies. Budget for me is 3* hotel or airbnb or bed and breakfast type of accommodation. We like to visit Sorrento, Positano and Amalfi. At the moment I am just looking at places of interest including a ferry to Capri. We are not beach people rather, walkers and hikers. Then we think of going to Sicily for a few days and just chose Palermo to start off with because of its history and architecture. After Palermo we have to make our way to Rome via Naples to start our 18 day Italy tour on 10th October. Your comments and suggestions are much appreciated.

Note you can fly from Naples to Palermo and then back to Rome, but I would think first about if you want to do that--it's a lot of travel for maybe not enough payoff (but it may be worth it to you!).
Does the tour include Naples? If not, those four days could easily be filled there, then hop on a train to Rome to start your tour.
The star rating does not mean a lot when it comes to prices--just go to a website like Booking.com to get a sense of prices (you can then book directly if you prefer). Even the exclusive Amalfi Coast has a range of prices.