John,
I've been to Italy several times. My family is actually from an area between Rome and the Amalfi coast. When travelling from Rome to Amalfi I would follow the recommendations from the other guys above, rent a vehicle, the roads are marked very well and overall it is very easy to get to Amalfi from Rome. Now I'm not going to sugar coat the driving habits, most people drive very aggressively and fast. In addition the road through the Amalfi coast is pretty exciting and nerve racking, the roads feel narrow especially when you have these large tour buses travelling the same road. When you are driving through the coast, the towns I would definitely hit are Sorrento, Positano, Amalfi and Ravello. If you plan on staying a few nights on the coast Ravello will be the most expensive, try staying in the other three towns, there are a lot of family run bed and breakfasts you can stay at. Do a web search and read the reviews online to find the one that best fits your needs. I have no recommendations where to stay because when I go, I go for the day and return to my parent's home. Now if you decide to drive and head up to Ravello,which is a must must must see, the road up can be fun or not so fun depending how comfortable you are driving up narrow roads where 2 cars barely fit.
If you want to try some other places I highly suggest Gaeta and Terracina. These are both beach towns that are great places to spend some time, just keep in mind that most of Europe is visiting these areas at the same time in August, normally the first 15 days of August. There is a fantastic beach called Bahia Blanca, they have a really nice setup with great food on site. Also if go to Gaeta there is an attraction called La Montagna Spaccata or The Cracked Mountain, do a web search.
Another area I would make an effort to hit is the island of Capri, this island will blow you away. Now with Rome, I would follow what others have said and not drive within the city, I do but I would not recommend it, it is very easy to get around the city by walking or by taking the public transit. In Rome you want to hit Trevi fountain, the Pantheon, the Collosieum, and the Vatican. When you go to the Vatican make sure you time it right because the Sistine chapel does not allow ticket purchases after 4:00 PM. Now if you have the time or make a second trip ;D make sure you hit Cinque Terre, it is another coastal town north of Rome comparable with the Amalfi.
When it comes to food and drink the prices are very reasonable even in the tourist areas. If you are driving make sure you stop at the Autogrill, this is Italy's version of a rest area, the food is really good. You get everything from a panini with prosciutto or a plate of pasta cooked to order.
If you have any questions let me know.
Carlo