Rome to Monte Cassino, then to Positano

Yes, the train station/city is Cassino, the abbey of Monte Cassino is above the town. There are trains on Sunday every 1.25 hours or so, the trip is a little more than 1.5 hours. I checked for 2/9, a Sunday, since the train is a Regionale and information is not posted more than 7 days ahead. If anything, trains in summer will be more frequent. One-way fare is about 7 euro.

There are frequent direct trains from Cassino to Napoli Centrale, where you can change to the Circumvesuviana (downstairs at Napoli Centrale, separate ticket window before you get to the staircase). Trains leave about every half hour for Sorrento, the last stop on the line. At Sorrento, walk outside the Circumvesuviana station and you will see buses - take the bus marked "Amalfi", it will stop in Positano (buses run about every half hour, the ride is almost an hour). You can also arrange for a car and driver from Napoli Centrale or from Sorrento.

Thank you, very helpful!
We like to avoid Naples if possible. Trying to figure out if there is a train from MC to Salerno, where we will get a bus to Amalfi and Positano.

There is nothing wrong with Naples.
It is possible to proceed from Naples to Salerno, instead of going to Sorrento and then to Positano. Then in Salerno you could take a bus to Amalfi and from Amalfi take another bus further north to Positano. It's longer that way, but you get to ride the bus on the whole Amalfi Coast road and see more of it that way. There are also boat options available from Salerno (and also Naples), however in October they may not be in operation.

Check trenitalia.com for alternatives to Naples (although I don't know why it should be avoided, especially just to change trains). Click on the British flag icon to translate the page into English. Remember that Regionale trains can only be viewed 7 days in advance of travel.