Transportation from Cinque Terre to Amalfi Coast.

Train and then bus e.g

Train
07:09 La Spezia Centrale
10:05 Roma Termini (Rome)
FA 8589
2nd 1h50 to change stations
11:55 Roma Tiburtina (Rome)
14:16 Salerno
https://www.trainline.eu/train-times/la-spezia-centrale-to-vietri-sul-mare-amalfi
Bus:
Salerno to Amalfi
http://www.sitasudtrasporti.it/

Which town in CT, and which town on AC? Makes a difference.

CT: Montorosso to AC: Amalfi

www.trenitalia.com
choose English flag for English language
Enter from MONTEROSSO to SALERNO
enter a date. If your date of travel is far into the future, just enter a day next week, the timetable won’t change much if at all.
Enter a time. To see trains from very early in the morning enter a very early hour, like 00 or 04.
I see a good fast travel combination departing Monterosso at 9:06am, via Pisa, Firenze SMN (Florence), then direct to Salerno (2 changes only, at Pisa and Florence). It’s on the fastest trains. Full price would be almost 100€ in standard (2nd) class, but you can save a bunch by buying ahead on line, if you can commit to a non refundable Supereconomy or Economy fare.

Quite frankly there is no easy way and no fast way for that journey. Roberto's suggestion of the 9.06 train seems best based on current schedules. It's still nearly 7 hours until you get to Salerno at 3.52 pm. Then you have to take the bus to Amalfi. The current SITA schedule shows hourly buses on the half hour (every two hours on weekends). The journey is 1.25 hours. That is the winter schedule. If you're traveling in summer, there will probably be more buses, but the journey could be slower due to heavier traffic. My guess is that you have to keep your luggage with you on the bus. Depending on when you're going to be there, you may be able to take a ferry, probably faster. It's an easy 10 minute walk from the train station to the pier.

"Quite frankly there is no easy way and no fast way for that journey."

Chani beat me to it. This is very important to understand, so you don't get too frustrated. It's going to be most of a day, and require several different trains/buses. As long as this is fine with you, it can work; just don't expect it to be something else.

In California it would be like going from Santa Barbara to Lake Tahoe. Also about 8 hours drive.
At least in Italy you have a decent train option, which is even faster than driving.
From Santa Barbara to Truckee, near Lake Tahoe, is a 16 hour trip on Amtrak.

There is also the option to fly. Monterosso is about halfway between Pisa and Genoa airports (about 1.5-2 hours from each). I don’t think there are no stop flights to Naples from Pisa, but there are from Genoa. Once you fly to Naples, NAP is about 1.5 hr from Amalfi via car and a bit longer via bus. However once you add, the airport transfers time, the airport check in and security procedures, and the flight time, the high speed trains+bus/ferry from Salerno to Amalfi, is just as fast door to door.
But if you want to explore the flight option you can.
www.trenitalia.com will give you the train timetable from Monterosso to Genova Piazza Principe or Genova Brignole (two main stations in Genoa). The train journey varies from 1h20min to over 2hr depending on the train.
From Genova (either station) to the airport there is a shuttle bus or taxis that can take you to the airport in less than 1/2 hour. You need to be at the airport at least 1 hr prior to departure.
Direct flights from Genova (GOA) to Naples (NAP) are operated by Volotea. Currently they have two dailies departing at 7am (a bit too early to get there from Monterosso on time) and 7pm (which however gets you to NAP in the evening at almost 8:30pm.
If you go in summer there may be more options, maybe also with other airlines like Alitalia. Check skyscanner.com.
The flight time is 1h20min.
Once in Naples the best option is probably a private car service (limo) to Amalfi. It should cost about 100€. There is also the option of a bus to Sorrento then another bus to Amalfi, but by then you may want the comfort of a car.
Looking at the current flight schedule, You could leave at 4:06pm from Monterosso station, get to Genova Brignole at 5:15pm. From there you can take the shuttle or taxi and be at the airport on time for the 7pm Volotea flight to NAP. Arrival at NAP is at 8:20pm. A private car service from NAP airport can get you to Amalfi by around 10pm.
Just over 6hrs door to door. Not too bad.
That’s the best we can do, my friend.