You're in luck. Cinisi is very close to the airport--about 6 miles away. It would be much shorter if you could walk across the runways! I assume a taxi wouldn't be too expensive, but I'd try to find out ahead of time approximately what that should cost, taxi drivers at airports being not always honest.
If you Google Cinisi you'll get a lot of hits. Perhaps one of them will give you a good sense of whether you need a car to get around. My guess is that it will not be necessary in the town itself, because it's a fairly small place, but of course if you want to explore the surrounding areas, a car would be cheaper than multiple taxis.
To get to Palermo after your visit to Cinisi, I suspect it will be best to taxi back to the airport. Google Maps is offering a variety of options from Cinisi itself via a combination of walking (to Cinisi Terrasini) and mixed modes of transportation, but they look nightmarish with luggage.
Renting a car at the airport and later using it to drive to Palermo should be done only after careful research. I would not want to drive in Palermo. No, indeed. Selecting a sane drop-off location would be essential.
For your lodgings you can take a look on booking.com. The Map View shows several options in the town, plus many more near the coast at Cinisi Terrasini. As always, read all the English-language reviews. I'm not sure you'll find many people on this forum who've stayed in a Cinisi hotel.