jenna,
Your original route of Milan > Cinque Terre > Rome > Amalfi Coast is a good one to work with. As someone else has suggested, using open-jaw flights will be the most cost effective and will save you time, which is important with such a very short time frame. You could also look at inbound flights to Pisa or Florence rather than Milan, and the outbound flight either from Naples or Rome (depending on which one offers the best price).
It would help to know where you're flying from, as one of the forum members may have some good tips on flights from your area.
I'd suggest going directly to the Cinque Terre, although that will make for a long travel day. Plan to spend at least 3 nights there, before moving on to Rome. That will be a fantastic place to recover from jet lag and have a "gentle" introduction to Italy (the culture becomes more intense as you head south). You'll first have to decide with of the five towns to stay in, and all of us here have our own preferences.
Travel by train from the Cinque Terre to Rome is very easy (the trip is about four hours, with at least one change at La Spezia Centrale). You can find all the train solutions on the Trenitalia website. Travel from Rome to Naples is also very easy on either the Trenitalia or Italo high speed trains.
Where are you planning to stay on the Amalfi Coast? Transportation from Naples will vary, depending on where you're going. Sorrento makes a good home base for that area, as it has reasonably good transportation connections (including an easy Bus ride direct to the Naples airport).
You may find it helpful to buy a copy of the RS Italy guidebook, as that has an enormous amount of information which will help plan not only sightseeing but also hotels, restaurants, transportation, etc. On the topic of "transportation", there are some potentially expensive "caveats" when using trains and other public transit in Italy, so you'll need to do some research on that. If you need more information, post another note.
Good luck with your planning!