I have driven the Amalfi coast road from Castellammare to Salerno...but the last time we were there we deliberately booked in late October/early November. It was a thrilling drive (on more than one day), but I am an enthusiast who loves those types of roads and I am very comfortable driving in Italy and elsewhere in Europe. Being truly off-season, some of the tourist places, restaurants and hotels were closing for the season. But the weather ended up being relatively sunny and warm (15 to 20C) with little or no rain.
The tour buses were gone, the kamikaze kids on their scooters and motorbikes were back at school, the line-ups for restaurants were non-existent and most of the people we met were locals. There were no problems finding a place to park in Sorrento, Positano, Amalfi or Ravello, and we had no problem finding a place to pull over along the coastal road at the many vista points. In short, we loved it. It was very relaxing.
We also prefer to stay in one place for a while and use it as a base for exploring the area. Because we had a car we chose to stay at what was then the Crowne Plaza Towers Hotel between Castellammare di Stabia and Vico Equense on the road to Sorrento. We picked it because it had a parking garage built into the mountain and beautiful views of Mount Vesuvius across the Bay of Naples. Because it was off-season we were able to stay in a large suite in a 4-star hotel for what we would pay for a small hotel room in a 2- or 3-star hotel in July or August. From there it was easy to do trips to Sorrento, Amalfi, Positano, Ravello and Pompeii. However, we could not go into the water and some tourist sites were closed, such as the boat tours in Capri.
Since you will be going in May or June, it will be more crowded, but with gorgeous weather and everything will be open. However, it should not have the intense crowds or heat of July and August. The coast road will be busy, so I do not recommend driving a car. I agree with those suggesting taking the train from Rome. We have travelled by train in Italy a lot and enjoy it. The tour buses will be operating on the coast road, so that will be a much safer and more relaxing way to take in the beautiful scenery of the Amalfi coast. Sorrento is the biggest town on the Amalfi coast, with lots of restaurants within walking distance, but the views from some of the hotels in Positano and Amalfi itself cannot be beat, but you will pay for that in peak season.
If you wish to tour around Tuscany later, then take the train to Florence and pick up a rental car there. Or use Florence as a base and just rent a car for the day to do day trips into the Tuscan countryside, although much of it is accessible by tour bus, too. You could also just take the train back to Rome from Sorrento and pick up a rental car in Rome and drive to Borgo al Vigne. And then return the car to Rome at the end of your trip. Just be aware of the ZTL (Zona a Traffico Limitato) zones in the centre of many Italian towns and cities. The fines are hefty if you enter them, particularly in a rental car. Enjoy your trip!!