Milan to Cinque Terre to Florence to Rome via train

Taking the train is not a problem. You will have no problems buying ticket the day of or day or two before because Italian trains rarely, if ever, sell out. If you purchase in advance you might be able to take advantage of the mini fares but that locks you into a schedule with no refundable tickets. Also some of the route will be on regional trains which have no reservations and no discounts so those routes cannot be purchased in advance other than at the station. Bahn.de give the schedules and may be a little easier to use than trenitalia.com.

You can easily go to the train station at the Malpensa airport and buy your train tickets straight through to the CT. If you are going to Monterosso, you could get lucky and get one of the IC train runs from Milan to Monterosso that require no train changes. You will change trains at Milan Centrale from the airport train. If you are going to other CT towns, you will most likely have to change trains in Monterosso for these other towns. Except for the IC train on which you will need seat reservations, all other trains on this overall run will be on Regionale trains. No seat reservations permitted. It's open seating. From the CT to Florence, this will also most likely be on Regionale trains although there are a few Eurostars if you have to change trains at Pisa. Your train fares will be really inexpensive up to this point. You won't get to to the premium high-speeds until you travel from Florence to Rome. There are about 55 high-speeds a day from Florence to Rome (2-3 per hour during the day). Each holds about 400 passengers. You won't have trouble getting tickets a day or two in advance of your travel.

Thanks very much! I should have mentioned that we're flying into Linate. Any advice on getting from Linate to Milano Stazione Centrale?

Take the bus.