Practicalities first. If you leave home on the 9th (Thurs), you get to Rome on the 10th (Fri) which is Easter weekend. Expect crowds everywhere. It's a very popular time for tourists. It's also less than 3 months away. The very first thing I'd do is see what lodging is available! The bargains go quickly. You may have slim pickings if you are watching your budget, so that may determine your options. The Amalfi Coast towns range from moderate to very expensive, and around Easter is probably high season prices in any event. Overall Sorrento is less expensive and Salerno less than Sorrento and Naples the least expensive.
Gerri suggests staying in 3 places the first week. That's quite a lot of moving around when you start out zonked. If you haven't been to Europe before, you won't know how much or how long jetlag will affect you. Coming from Iowa, you are going to have at least one plane transfer (do you have your flights yet?). Some people sleep well on overnight flights, many don't. I would limit it to 2 bases, Salerno for sure (for Paestum and the AC) and then either Sorrento or Naples for the Roman excavations and Capri). Naples has interesting sights and the archaeology museum is a wow. Sorrento is charming but has no sights. The Circumvesuviana will get you to Herculaneum, Pompeii from Naples or Sorrento, not much difference. I believe there are ferries to Capri from both. Naples is an acquired taste and is not for the faint-hearted, but she rewards the intrepid visitor.
If you'd been to Italy before, I'd recommend your starting with a few days in Naples, then Salerno, but Naples is a poor choice as an introduction to Italy. Salerno is definitely easier to get to than Sorrento - train from the airport to Termini (Rome) and get train to Salerno. Stay at least 3N. Friday is shot - you'll be exhausted by the time you get there and probably just struggling to stay awake until dark. Paestum is about a half-day (the bus may be better than the train). Make sure the Archaeology Museum is open on the day you plan to go. The temples are best seen in the late afternoon sunlight, sunset and then illuminated. The bad news is that in April sunset isn't until around 7.30. Salerno itself has an interesting medieval center to explore. The ferries to the AC start in Salerno, as do the buses. There are several ferry lines, SITA is the bus company. There are 3 bus lines, Salerno-Amalfi, Amalfi-Ravello, and Amalfi-Sorrento. The scenic stretch of the AC is pretty much from Positano to (almost) Salerno. You may be able to reserve ferries in advance. For buses, you just have to stand in line. They usually run about every 30 minutes and when there are crowds they fill up. For me, the AC towns are unremarkable to visit (lovely to look at from afar), the joy of the AC is the scenery.
I have no idea what it's like on the AC over Easter. Perhaps some folks here can give you more info. I'd try to do some online research by googling and using the tripadvisor.com forums which get a lot of locals.