We will be arriving by Delta to Milan -Should we purchase a train ticket before we arrive? With airline delays not sure?
Just buy tickets when you arrive. The savings are usually not commensurate with the risk, IMO.
I'm Assuming you'll be arriving at MXP and not at Linate or Bergamo. The Malpensa Express airport train from MXP to Milano Centrale has no reserved seats, so it can't sell out.
Trenitalia runs direct Intercity trains from Milano Centrale to Monterosso that have only reserved seats. During the summer week-ends and in August these trains sometimes sell out as the Milanese flee to the seaside; at other times you can always get a couple of seats at the last minute. Of course you'll have to pay the full BASE fare.
Most trains from Monterosso to the other Cinque Terre villages are local Regionale that, again, can't sell out.
You can purchase a through ticket from Malpensa to your final destination in the Cinque Terre at the airport counters; there is no need to queue again at Milano Centrale.
Do not forget to time-stamp locally purchased Regionale tickets (those without a reserved seat) before getting on.