dgame,
A few thoughts on your questions.....
Which of the five Cinque Terre towns are you staying in?
You'll arrive at Malpensa Terminal 2 and since that's your first entry to the Schengen area you'll have to go through Passport control, where your Passports will be stamped with the entry date. I don't remember whether checked luggage is retrieved before or after Passport control, but the time will be about the same. I assume you'll have nothing to declare at Customs, so that will be a walk-through. The time at Passport control will depend on how many agents they have on duty and how many flights arrive at the same time, but it shouldn't be excessively long (of course in Italy, one can never tell).
You'll then make your way to the station for the the Malpensa Express (train) from there to Milano Centrale, which will take 58 minutes. You can buy your ticket at the airport but be sure to validate the ticket prior to boarding the train, or you'll risk hefty fines! The time and specifics for the train to the Cinque Terre will depend on which train you connect with. Using an arbitrary date of 8 June and Monterosso as the destination station, the 14:37 will be the quickest and easiest as it's an Intercity direct. I suppose there's a slight possibility you could connect with that one, but I wouldn't recommend buying an advance ticket as if you miss the train, the ticket is worthless and you'd have to buy another at full (Base) fare.
If you decide to buy tickets online, I find that www.trainline.eu is a very easy site to use. Registration is required along with payment details but once that's done, the process is very easy. They also have an App. They sell tickets for not only Trenitalia but also Italo and other networks in Europe, so that provides "one stop shopping".