I think you'll love Palermo, and with only two weeks, you are wise to limit yourself to that area. You'll get lots of suggestions, but I'm going to mention one that I don't see here very often: Trapani. We used Trapani as a one of the bases on our last trip (for 9-11 days, I can't remember now), and were very happy there. However, it's an easy bus ride of about an hour from Palermo. You wouldn't need a car, and actually, and certainly in Palermo, it would be more of a headache than not. Trapani has everything you're looking for. If you've got some wiggle room in your Airbnb booking, I think you'd do well to reconsider a week there and a week in Palermo. Trapani is a small town, completely different from Palermo, and a great base for day trips. There is only one main street, but others that are ready for exploring. It has a small museum, a couple of unique churches (see the bronze gate at the Basilica of St. Lawrence and the wood/cork/paper mache sculptures used for Lenten services), a small beach (although there are others that are 'better' nearer to Palermo and Favignana), and the Ligny Tower. From there, we day tripped to the salt flats and the Baglio family's museum of salt, both well worth the excursion. (Public transportation is problematic but possible; a car would be much easier here, although we arranged for a driver and that worked great. It's not that far; if they have Uber, that would be a reasonable option.) We also visited Erice (take your coat; it's cold up there), Selinunte, Marsala and Mazara del Vallo from Trapani, but I think you can get to all of them from Palermo as well. I didn't research those transportation options. We enjoyed the island of Favignana, too. It has big deal beaches but the town is also interesting (and it does have a small beach near the tuna museum). Make sure to see the tuna museum if you go. Traditional tuna fishing was the mainstay of the economy for decades there. In April, I'm not sure what the ferry shuttle schedule will be, as it wouldn't be the same as high season, but check it carefully to make sure you won't be stranded out there overnight (although I understand there are nice hotels on the beach side of the island). RS recommended that you purchase two one-way tickets so as to give you some flexibility in return time, but in September we were glad we purchased the round-trip up front as all the end-of-day trips were sold out when we were returning. Instead of going independently, especially if you want to go to the beach, you might want to consider a package for the ferry and transportation to the beaches, which are a ride away from the port. You could also rent a bike on the pier if you wanted a little more adventure. I understand you can also get there from Palermo. Probably too far-flung from Palermo for you with your limited time, and a car or driver would be necessary, are Cretto di Burri and Poggioreale. There was a huge earthquake there in 1968 which destroyed both villages. Poggioreale is a spooky ghost town now. Cretto is interesting because the entire town of Gibellina, also destroyed, was encased in huge white cement "boxes" to create a maze of landscape artwork. You can wander the "town" as the streets were left as the paths. Thought provoking and fascinating. PM me if you want more details.
From Palermo in the opposite direction of course is Cefalu and Monreale, but San Vito Lo Capo is worth it too. It has gorgeous beaches, the Zingaro Nature Reserve, and boat trips and snorkeling. The food is amazing. Couscous is the highlight, although local tuna dishes are also great. Unfortunately, you'd miss the annual couscous festival/competition, but it's still 100% worth going. It is a schlep, though, about 2 hours from Palermo. If you decide to go, an overnight would probably be best there. Whatever direction you go, you can't go wrong. Sicily is wonderful.
