There is usually no extra charge for picking up and dropping off a car in different cities in Italy; there are heavy charges if you pick up in one country and drop in another.
As for companies, many here like to use AutoEurope, Gemut, and Kemwel. These are consolidators; your actual rental will be with a company such as Hertz, Europcar, Sixt, etc. The only way to know who offers the best deal is to get quotes from them all (both the consolidators and the car rental companies). I know of several people (and I'm one of them) who found that Kemwel had, by far, the best deal for their needs, but you have to check. Be sure your quote includes all the insurance you want (I insist on zero deductible; one company offered me a good rate, but with a two thousand euro deductible!)
As for where to pick up the car, if you are going from the Cinque Terre to Tuscany, La Spezia has the closest rentals to the Cinque Terre. You would drop the car at the airport before flying to Sicily.
For flying to Sicily, know that while Palermo is the bigger city (airport code PMO), Catania (CTA) has the bigger airport with more flights. Rome certainly has the most flight options to Sicily, but other Italian cities usually have 1 flight a day to Palermo and 1-2 flights a day to Catania. Going from Tuscany to Sicily, look at options from Florence (FLR) and Pisa (PSA). If you don't find what you want, you may prefer to drive to Rome (FCO), where there are numerous flights per day, on multiple airlines, to both CTA and PMO.
To find flights, use Skyscanner: https://www.skyscanner.com/. Note that many flights are on budget airlines; be sure to go their websites and do a dummy booking (up to where you put in your credit card) to learn of all the rules and fees.
It seems that Alitalia does not fly from Pisa or Florence to Sicily. But if you are flying from Rome, you will have a choice of carriers. Alitalia will allow one free checked bag and can have a more generous carry-on allowance, so it can work out cheaper than the budget carriers. If you are using Alitalia for these domestic flights, use their International website, which has the same prices as the Italian website but also has an English option: https://www.alitalia.com/en_en/ The US Alitalia website has higher fares - for the exact same flights!
