Civitavecchia to Praiano....what is fastest and cheapest way to go for 5 people

It is not simply or fast. Around 5 hours and 3 changes. See -- https://community.ricksteves.com/travel-forum/italy/travel-from-rome-to-praiano-train-bus --- The posting is old but the information is accurate but starts at the airport. It was copied by GAC from the Praiano web site. Or goggle Praiano, Italy for updated information. Go to trenitalia.com, select English, and plug in Civit.... to Salerno. That will give you the schedule. You would not have to go through Termini. But you will have to take a bus from Salerno to Praiano.

Thank you, for that information....I was wondering if there was a DIRECT train to Naples or Sorrento from Civitavecchia ....would prefer not to have to backtrack from Salerno to Praiano...husband has trouble lifting suitcases....is having hand surgery in 2 weeks, but should be good to go by June, July. Guess we could get water taxi (hydrofoil) to Amalfi, but don’t know how pricey that is either...

There are no direct trains. Direct trains are only between major metro areas. Most train travel will involve changes. You are not back tracking from Salerno. You need to pack light -- carryon only -- since this will involve some walking in addition to changes.

A rental car (or two) is the fastest and most convenient option. Possibly even not that expensive for 5 people.
The train option for 5 requires, multiple changes on various modes of transportation, all costing about €250 to €350 each way depending if the last leg is done by bus or taxi.
If you have two adult drivers, you could rent 2 economy cars for $150 each a week. Add fuel and parking costs and I bet it’s still cheaper than the train.
If renting a car is not for you, then I would take the train from Civitavecchia to Naples (nearly all options require a change in Rome). From Naples I would then book a private car transfer with a limo service (no more than €120 for a van ride to Praiano). The cost of this last option would be less than €300, each way, for all of you (€175 for the train, plus the car transfer). The total journey time would be comparable to driving yourself all the way from Civitavecchia.

You have chosen to visit a distant and slightly out-of-the-way place. That's one big reason why people go there. The term "backtracking" doesn't really apply to a place that has no trains of any kind. In fact, motor vehicle travel on the Amalfi Coast can be annoyingly crowded and slow from May to October, but you didn't say which month you are traveling.

You may need to take a taxi to the rail station in Civitavecchia, although you will still (most likely ... ) have to take the port's free bus from the ship to the entrance to the port. It's very likely that if you PAY (I used capitals because this usually is over $100 per person) for the ship's Disembarkation Bus to somewhere, it will board you beside the gangplank. Alas, many such trips to Rome include a tour that maybe you don't want to pay for. You want to go to the main rail station, Termini. Maybe you should find out how much the ship's private car service is, because it MIGHT come to the gangplank.

This leads me to the possiblity of asking your hotel who their "land travel partner" is, and ordering a car service with a driver, to pick you up at the Salerno train station. At least there is fast, efficient, affordable train direct from Rome to Salerno. But to save money, you have to buy non-changeable, non-refundable advance tickets. During some months, there might be a ferry from Salerno to Amalfi or Positano, but you'd still need local surface transport. I'm guessing you don't want to hump onto the really cheap SITA bus from Salerno. But you'd have to change busses (and handle your own luggage) in Amalfi. It's not that ususual for people with moderate travel budgets to have a car service pick them up in Naples and take them to Positano, so there's no reason not to ask the hotel concierge to price that car-and-driver trip.

One could say that Priano is closer to Sorrento, but you clearly don't want to use the rough-edged commuter rail service from Naples to Sorrento, rather than going direct to Salerno.

You would be correct if you said, "Gee, fast .... and cheap .... seem to be incompatible attributes." But Priano is so far that fast is a term that should be discarded. I get the idea that physical effort required is by far your most important consideration.

I've not taken it so can't vouch for its reliability but there is a shared shuttle bus to Praiano from Rome Termini or Napoli Centrale train stations. .

https://www.positanoshuttle.com/services/shuttle-bus/

But the other thing to consider is that you'll eventually be leaving Praiano for your next destination so may be facing the same time/cost issues. As the others have stated, for this location, "fast" and "cheap" don't generally apply to covering the distance from Civitavecchia, and could be the case from Praiano to other locations as well.

If none of the options provided above suit, would you be willing to change your location for one a little easier/quicker/more convenient to manage, assuming you can cancel your hotel reservations with no penalty? Also, it would help to know what month you'll be traveling, what you intend to do for sightseeing whilst visiting the A.C, and how long you'll be staying?

You are aware that there is no public ferry service to Praiano so the only public transport option there is SITA bus? You'd mentioned "water taxi" to Amalfi but those are not the high-speed hydrofoils used to move most visitors around the coast (and don't run to all locations in every season). This is what comes up under "water taxi" on the Praiano website:(http://www.praianonline.com/eng/informazioni_praiano.html):

https://www.positanoboats.info
https://www.positanoboats.info/transfers/

Definitely not an inexpensive option!

It is basically same distance from Naples to Praiano as it is from Salerno to Praiano 53 km vs 57 km- it's not a back track- just going in a diff direction.
Looks like you will have to take trains somewhere- Rome to Naples or Rome to Salerno unless you go private transfer the whole way
your choice

We used this transfer service and were very pleased with their rates and service
https://www.topexcursionsorrento.com/en/transfers
Scroll down for most of their prices
Quick look shows Naples to Praiano is 120€ vs Salerno to Praiano 137€- for up to 8 people

Transfer from Civitavecchia to Praiano looks to be 550€

You gotten a lot of good advice. I'll just add a few tips about Civitavecchia since I was there recently. There are two steps when you embark at the port. The first is a free shuttle bus from the dock to a public transit center called Largo della Pace. The second step is getting from Largo della Pace to the train station/Rome airport/other destination. If you plan on renting a car, there is a Europcar agency within easy walking distance of Largo della Pace. If you want to get to the train station, there is a convenient shuttle bus that will get you there quickly and cheaply (cabs overcharge, but with five people it might be worth it anyway). Buy your bus ticket BEFORE you board at the ticket kiosk--no tickets available on the bus. If you take the train to Rome Termini, be aware that most trains from Civitavecchia arrive at platforms far, far away from the mainline platforms. It really is a 20-minute walk. So factor that in when you plan your transfer to a mainline train. Have a great trip.