To explore the island and get where it is wanted, it is good to hire a car. Renting a car on vacation is common in tourist areas and sometimes on some Greek or Spanish islands, 99% of cars in car parks at tourist attractions are cars from rental companies.
You can hire a car hire in large global network (Herz, Avis, Goldcar) or on a local, smaller rentals on site. Surely it's not worth to lend a car on the Greek islands, or to book it on the internet few days before. The internet prices are fixed, mostly much higher than right on the spot. Especially that it's almost a necessity to bargain. Unless You're renting a car for one day - then it's better to get list price.
Personally I recommend bigger rentals, but local - such rentals on Zakynthos are, for example Faros, EuroHire, Ionian Cars - these are rentals, the size bigger than an island, but You probably won't meet them in other countries.
Generally: large rental companies have cars mostly with a little better quality (newer and higher-class), but they are much more expensive and corporate standards can include full insurance, which might cost twice that much then the same rental car without it. Frequently in the ordinary rental there are also excluded many things, like damage to the chassis, wheels, mirrors, windows etc.
It is better to use local rental that often gives better offers than the ones on the internet by offering full insurance. It is good to read carefully the conditions of hiring that we get from the rental, because the "full insurance" is often understood in each differently. For example, in EuroHire actually in the rental price called "full insurance" we had mentioned that damaged wheels or mirrors are our responsibility to fix.
There are many rental companies, each offering scooters, small quads (which are very common on Zakynthos) and cars. Zakynthos is dominated by small cars like the Kia Picanto, Citronen C1, VW Polo etc. Rentals offer larger cars as well, mostly "roadsters" - at the same price there was offered Suzuki Jimmy (with opened back) and Kia Sportage. Practically - larger cars are not worth renting - even if we want to drive to any place on the island - you can get there with the small cars - including gravel roads such as Dafni beach.
Renting a car is generally more expensive than, for example Canary Islands and the price for one day for a small car that initially got was 60 euros per day. Finally, when renting a car for 9 days we went down to the price of 400 euros.
Adventure with a car:
Kia Picanto from EuroHire - after driving approximately 10 km we figured out that on every hole something creaks in the front right wheel - it creaked from the very beginning, but initially on a straight road we didn't notice it. We went to the rental company - nice mister had driven with us to check this and after noticing that we were right, he complained that we have damaged a car by driving in to a hole and he kept complaining that it was our fault and so on. He told us to park the car by our hotel and to leave the keys in reception and the next day at 7 am there will be a new car waiting for us.
It's worth checking immediately after renting if everything is in order - whether if it's not a faulty one or not scratched - and show it immidietaly (or do photographs) - if it wasn't us we have a right to defend.
In smaller rental stores usually You give the car with the same amount of fuel as You had it when you rent it (which is generally convenient). In a larger companies there may be different rules, their own rules. For example the company may require from You buying fully fueled car (of course the price will be slightly larger than at the station)
The longer is the time you want the car, the cheaper it would probably be per day (You need to bargain).
Extras like a child seat (or so-called "Booster") are paid additionally, but asking about the car and talking about it right away, usually gets it for free as part of the negotiations.
Visiting everything on the island, what could be visited, having a car hired for nine days, we traveled less than 400 km. It is a small island :-)
Cars are decent, but because of the sea and salt water, everything in them most frequently creaks. Kia Picanto had a mileage around 60 thousand of km, and C1 Citrone which we had, had a mileage around 23 thousand of km.