Is it worth to rent a car?

Most people going to the Amalfi Coast go by train from Naples and use public transit (buses) going from place to place. Driving is difficult down there, and parking is apparently hard to find.
I never rent a car at an airport if it can be avoided, as there are always extra taxes and carrying fees.
Many rental companies prefer to rent to those 25 and older--especially the ones that have credit cards with large limits. They want to be assured that their cars will be fixed if the renter has an accident. Italy's not a place for a first time renter. And forget trying to rent one with a Visa ATM, as rental companies don't accept that usually.

I don't think you should rent the car, not due to costs but due to traffic. Italy's autostrade have tolls and relatively easy driving but the view roads that you'd be taking are overburdened with traffic and no place to park "wherever you want." Public transport works fine for everything you want to see. Buses along the Amalfi coast buses run frequently enough that you can make a stop and board the next bus, space permitting.

We have never driven in that part of Italy, but we have driven in many countries in Europe, including northern Italy. We have found Italy to be the most difficult to drive in. The highways are fine, even Tuscany is fine, but we have also been in total craziness, even dangerous driving where cars pass within your single lane, cars, people and bikes coming right into the roadway, and cars passing so close you think you will loose your side mirrors.

Especially as a young driver, I would avoid the stress of driving. I was happy to get out of Italy with our car, and hubby lived and drove in Germany for a few years, so he is experienced. It can be nuts!!

Jessi,

A car rental in that part of Italy is perhaps not the best idea. Driving in Naples is not for the faint-of-heart as normal traffic rules don't seem to apply there. The road along the coast is narrow in parts and can at times be thick with traffic, including large buses.

A few other things to keep in mind....

• For driving in Italy, each driver listed on the rental form must have the compulsory International Driver's Permit, which is used in conjunction with your home D.L. These are valid for one year, and easily obtained at any CAA/AAA office (two Passport-sized photos required, which may be provided by the issuing office). Failure to produce an IDP if requested can result in fines on the spot!
• You may also want to have a look at some of the other posts here concerning the dreaded Zona Traffico Limitato areas that are becoming increasingly prevalent in many Italian towns & cities. EACH PASS through one of these areas will result in a hefty fine, which you won't know about until several months after you return home!
• There's also the issue of parking tickets, high fuel costs, tolls and automated speed cameras including the devious Traffic Tutor system which measures not only instantaneous speed but also average between two points. Violate either or both parameter and expensive tickets will follow. You may also encounter automated fuel pumps that will only accept Chip & Pin credit cards.
• A GPS along with a good Map would also be prudent. Be sure to give some thought to the question of CDW and theft insurance. In Italy, buying CDW from the rental firm is usually compulsory, so coverage provided by credit cards will not apply.

While a car will provide more flexibility, you might find that well planned travel by public transit is an easier and less problematic option.

Capri is an island so cannot take your car there, I assume you know that
Assume you will pay for your car to be parked while you are in Capri
The amalfi coast towns you mention are all served by constant Sita buses, there is no schedule to worry about they come every 15 minutes or so well into the night
Does not get you to Salerno but does the other towns you mention
Main issue with the car is parking, it will not be free and could end up a long walk away from where you want to be in Positano, Amalfi, Sorrento
By the time you add 25 euro per day parking you costs will go up considerably
Only reason a car makes sense or would be remotely convenient is if you are staying somewhere not served by the bus route
There is no way a car will be cheaper and I don't think it will be as liberating to your schedule as you think
I think the Amalfi coast roads can be fun though a bit scary and if you are driving you cannot enjoy the view. The view from the bus is glorious
Naples driving is insane you definitely do not want to drive anywhere outside of the airport there!!!

The bus does get you to Salerno, you change buses in Amalfi. You can get to Paestum from Salerno by train as well.

As a solo traveler, I don't find renting a car worth the money, or the parking hassle in congested areas.

My rule of thumb on driving in Italy is: if the train goes where you want to go, take the train.