Day trips from Palermo
We took two day trips from Palermo and we had planned on taking some more, but the trains on the Southern coast of Sicily were heavily damaged the first week we were here during some storms. So we only made it to Cefalu and Mondello - both we can highly recommend.
Cefalu is about an hour or so from Palermo and there are about hourly trains - both reserve seating and general boarding. I would recommend getting the reserve seating one — especially if you are here in the summer! Even in the winter, it was a very crowded train back on the non-reserve seating train. There is a lovely boardwalk and a nice hike up the hill (costs 5 euro per person) - there are no facilities on top, so make sure you take water etc. We had picked up sandwiches at a small deli and hiked up to the Temple of Diana. This is noted as an easy route, but it is a ton of stairs, but only about halfway up the entire route. We stopped a few times to take in the sights and catch our breath.
Mondello is a bus ride away and honestly, if we come back to Sicily, we will likely stay here. It is essentially a Palermo suburb, but has a nice beach and boardwalk. The bus route was great on the way back, but was detoured due to a bike race (may happen each Sunday morning) - I recommend waiting until the afternoon, as I was very car sick by the time we got there and it took an extra 20 minutes. There was a cute kids parade for Carnival. The small grocery store on the boardwalk was perfect to pick up a sandwich and drinks for a little picnic.