Milan to Cinque Terra

The website of the Italian railways http://www.trenitalia.com/tcom-en gives all the connections (you might have to use the Italian originals of the places names).
For the fast trains advance buying might give you considerable discount. The drawback is that those tickets are tied to a specific train at a specific hour on a specific day, and are not refundable. If you plan to take a train directly after your flight, you run the risk of missing that train and being left with a worthless ticket.

I am looking for the cheapest option to get from Milan airport to any of the Cinque Terra towns via rail.

Which airport, Malpensa or Linate?
The schedules for after June 12 are still not posted yet, so keep checking. I did see SuperEconomy tickets available for 9 EUR on direct (to Monterosso) IC trains in early June

You'll see that these trains are relatively cheap even at full fare, if locking in a specific departure online seems either risky or difficult.

LeAnna,

The first thing you'll have to do is travel from Milan / MXP to Milano Centrale. I'm assuming you're arriving at Terminal 1 after a flight from the U.S.?

The easiest way to get into Milan is the Malpensa Express which makes the trip in about 51 minutes. You can buy your tickets at the airport rail station (and possibly your tickets for the ongoing trip as well). A few important points to note. First, the M.E. has two routes, one going to Milano Centrale and one to Milano Cadorna. Be sure to board the correct train! Next, you MUST validate (time & date stamp) your tickets prior to boarding the train or risk hefty fines.

You'll arrive on track (Binario) 1 or 2 at Milano Centrale, and will need to walk into the main hall to connect with your next train. It's difficult to recommend any specific trains or routes, as the easiest solution is just to connect with the next train heading to the Cinque Terre. There are several direct (no changes) departures each day from Milan to Monterosso using Intercity trains, and that's the easiest solution (especially after a long international flight). Those trains have compulsory seat reservations, which will be provided with your ticket. There are numerous other solutions using various types of trains, with varying travel times and numbers of changes.

What time does your flight arrive?