Getting around the Amalfi Coast: car, bus, or train?

Also, I should add that we will be in Italy from May 22-June 1. I expect the Amalfi Coast tourist season will be picking up, so I'm wondering whether the roads will be packed with people and buses at that time or not? My biggest concern is the cost of hiring a driver from Naples to Praiano and back, there will be four of us total. We'd like to save money if possible.

Since you are staying in Praiano, a rental car would be my choice. However only if you are a skilled driver. The Amalfi road is narrow and curvy. Also traffic in Naples puts Manhattan to shame and compared to how Neapolitans drive, NYC cab drivers are courteous old ladies. Parking on the Amalfi coast comes to a premium (about 25 euro daily), but inquire with your hotel.
You should rent your car downtown Rome, since you are there already. Renting from the airports is significantly more expensive. There are rental office at Termini station and Tiburtina station (or nearby) and also other locations in the city, depending on the company you use.
If you don't feel comfortable renting a car and driving inside a city like Rome, then obviously you shouldn't drive at all. If driving in Rome is intimidating, don't even try in Naples.

Thanks Robert! We're renting a house through AirBnB, and there is a parking space available for us. I've driven around windy, very narrow roads in Ireland and New Zealand - is the Amalfi road similar?

Thanks for the tip about renting in downtown Rome! I'm surprised renting is more expensive at the Roman airports, because in many places where we've traveled it's more expensive to rent a car downtown. This is very good to know. If we rent a car, we wouldn't need to go to Naples, really - we would stop in Pompeii for a few hours and continue to Praiano. From looking at a map, we could bypass Naples by taking the A-30.

Thank you Roberto for being one of the few people on this forum not to completely trash the idea of driving the Amalfi Coast. Since you are now in California you may be familiar with California Route 1 through Big Sur and/or the Tioga Pass Road into the east side of Yosemite. I have driven a motor home towing a car (think 50 ft total length) over these two roads and have assumed I could handle the Amalfi Coast in mid May.

Compared to other parts of Italy and Europe, this was the most frustrating driving I have done: Without much plan, I drove the Amalfi Coast westbound, with the setting sun in my eyes, and survived it. But due to the heavy traffic, I can't say that I enjoyed it. I think traffic will be the main difference between this route and the Californian routes Bob mentioned. Similarly, when I accidentally got off the autoroute one stop too soon before Herculaneum, I found noontime traffic through the neighboring towns to be doing "every wrong thing" they could think of, which required a lot of patience.

Just don't be in a hurry to get anywhere - relax - take your time and enjoy the ride. You'll be fine.